Content-Type: multipart/mixed; boundary="-------------0110240829350" This is a multi-part message in MIME format. ---------------0110240829350 Content-Type: text/plain; name="01-396.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-396.keywords" Differential Equations ---------------0110240829350 Content-Type: application/postscript; name="absMP.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="absMP.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: arXiv:math-ph/0110029 v1 24 Oct 2001 %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -z -R -K1 abs.dvi -o %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2001.10.24:1315 %%BeginProcSet: texc.pro /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul 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Fl(X)727 912 y Fk(j)r Fs(=)p Fk(m)810 854 y Fo(s)833 861 y Fk(m;j)892 854 y Fp(\()p Fo(a)937 861 y Fs(1)957 854 y Fo(;)8 b(a)1005 861 y Fs(2)1024 854 y Fo(;)g(:)g(:)g(:)15 b(;)8 b(a)1167 861 y Fk(j)1185 854 y Fp(\))g Fo(a)1238 861 y Fk(k)q Fr(\000)p Fk(j)1352 854 y Fp(for)16 b Fo(m)11 b Fp(+)g(1)j Fn(\024)g Fo(k)r(:)129 997 y Fp(In)i(the)g(space)g(of)h (formal)e(p)q(o)o(w)o(er)h(series,)f Fm(R)p Fp([[)o Fo(x)p Fp(])o(],)e(it)j(holds)287 1050 y Fl( )345 1073 y Fr(1)326 1088 y Fl(X)330 1194 y Fk(k)q Fs(=1)406 1135 y Fo(a)432 1142 y Fk(k)453 1135 y Fo(x)481 1114 y Fk(k)502 1050 y Fl(!)542 1061 y Fk(m)589 1135 y Fp(=)662 1073 y Fr(1)644 1088 y Fl(X)641 1194 y Fk(k)q Fs(=)p Fk(m)727 1135 y Fo(s)750 1142 y Fk(m;k)812 1135 y Fp(\()p Fo(a)857 1142 y Fs(1)877 1135 y Fo(;)8 b(a)925 1142 y Fs(2)944 1135 y Fo(;)g(:)g(:)g(:)15 b(;)8 b(a)1087 1142 y Fk(k)1108 1135 y Fp(\))g Fo(x)1163 1114 y Fk(k)1185 1135 y Fo(;)56 b(m)14 b Fp(=)f(0)p Fo(;)8 b Fp(1)p Fo(;)g Fp(2)p Fo(;)g(:)g(:)g(:)17 b(:)129 1270 y Fp(This)f(implies)e(that)i(if)561 1398 y Fo(a)d Fp(=)671 1336 y Fr(1)652 1351 y Fl(X)656 1457 y Fk(k)q Fs(=1)732 1398 y Fo(a)758 1405 y Fk(k)788 1398 y Fo(x)816 1378 y Fk(k)837 1398 y Fo(;)8 b(f)19 b Fp(=)972 1336 y Fr(1)954 1351 y Fl(X)957 1457 y Fk(k)q Fs(=0)1034 1398 y Fo(f)1058 1405 y Fk(k)1087 1398 y Fo(x)1115 1378 y Fk(k)1150 1398 y Fn(2)14 b Fm(R)p Fp([[)o Fo(x)p Fp(]])129 1534 y(then)298 1599 y Fr(1)280 1614 y Fl(X)278 1718 y Fk(m)p Fs(=0)362 1661 y Fo(f)386 1668 y Fk(m)428 1661 y Fo(a)454 1640 y Fk(m)501 1661 y Fp(=)571 1599 y Fr(1)552 1614 y Fl(X)556 1720 y Fk(k)q Fs(=0)633 1661 y Fo(g)656 1668 y Fk(k)686 1661 y Fo(x)714 1640 y Fk(k)783 1661 y Fp(where)i Fo(g)947 1668 y Fk(k)983 1661 y Fp(=)1063 1599 y Fk(k)1037 1614 y Fl(X)1035 1718 y Fk(m)p Fs(=0)1119 1661 y Fo(f)1143 1668 y Fk(m)1177 1661 y Fo(s)1200 1668 y Fk(m;k)1262 1661 y Fp(\()p Fo(a)1307 1668 y Fs(1)1326 1661 y Fo(;)8 b(a)1374 1668 y Fs(2)1393 1661 y Fo(;)g(:)g(:)g(:)16 b(;)8 b(a)1537 1668 y Fk(k)1558 1661 y Fp(\))g Fo(:)86 b Fp(\(3\))202 1796 y(F)l(urthermore,)14 b(set)400 1933 y Fo(\033)430 1912 y Fs(0)428 1945 y Fk(k)449 1933 y Fp(\()p Fo(a)494 1940 y Fs(1)514 1933 y Fo(;)8 b(a)562 1940 y Fs(2)581 1933 y Fo(;)g(:)g(:)g(:)16 b(;)8 b(a)725 1940 y Fk(k)745 1933 y Fp(\))14 b(=)856 1871 y Fk(k)830 1886 y Fl(X)835 1991 y Fk(j)r Fs(=1)915 1899 y Fp(\()p Fn(\000)p Fp(1\))1016 1881 y Fk(j)r Fs(+1)p 915 1922 165 2 v 986 1967 a Fo(j)1093 1933 y(s)1116 1940 y Fk(j;k)1161 1933 y Fp(\()p Fo(a)1206 1940 y Fs(1)1225 1933 y Fo(;)8 b(a)1273 1940 y Fs(2)1293 1933 y Fo(;)g(:)g(:)g(:)15 b(;)8 b(a)1436 1940 y Fk(k)1457 1933 y Fp(\))209 b(\(4\))129 2074 y(and)268 2201 y Fo(\033)298 2180 y Fk(m)296 2213 y(k)331 2201 y Fp(\()p Fo(a)376 2208 y Fs(1)395 2201 y Fo(;)8 b(a)443 2208 y Fs(2)462 2201 y Fo(;)g(:)g(:)g(:)16 b(;)8 b(a)606 2208 y Fk(k)627 2201 y Fp(\))14 b(=)738 2139 y Fk(k)711 2154 y Fl(X)717 2259 y Fk(j)r Fs(=0)792 2131 y Fl(\022)828 2167 y Fn(\000)p Fo(m)858 2235 y(j)910 2131 y Fl(\023)955 2201 y Fo(s)978 2208 y Fk(j;k)1023 2201 y Fp(\()p Fo(a)1068 2208 y Fs(1)1087 2201 y Fo(;)8 b(a)1135 2208 y Fs(2)1154 2201 y Fo(;)g(:)g(:)g(:)16 b(;)8 b(a)1298 2208 y Fk(k)1319 2201 y Fp(\))49 b(for)16 b Fo(m)e Fn(\025)f Fp(1)p Fo(:)77 b Fp(\(5\))129 2344 y(Then)16 b(it)g(holds,)g(in)g Fm(R)p Fp([)o([)p Fo(x)p Fp(])o(],)254 2481 y(ln)303 2396 y Fl( )343 2481 y Fp(1)11 b(+)445 2419 y Fr(1)427 2433 y Fl(X)431 2540 y Fk(k)q Fs(=1)507 2481 y Fo(a)533 2488 y Fk(k)554 2481 y Fo(x)582 2460 y Fk(k)603 2396 y Fl(!)684 2481 y Fp(=)782 2419 y Fr(1)764 2433 y Fl(X)768 2540 y Fk(k)q Fs(=1)844 2481 y Fo(\033)874 2460 y Fs(0)872 2493 y Fk(k)893 2481 y Fp(\()p Fo(a)938 2488 y Fs(1)958 2481 y Fo(;)d(a)1006 2488 y Fs(2)1025 2481 y Fo(;)g(:)g(:)g(:)15 b(;)8 b(a)1168 2488 y Fk(k)1189 2481 y Fp(\))g Fo(x)1244 2460 y Fk(k)1265 2481 y Fo(;)242 2570 y Fl( )282 2655 y Fp(1)j(+)385 2593 y Fr(1)366 2608 y Fl(X)370 2714 y Fk(k)q Fs(=1)447 2655 y Fo(a)473 2662 y Fk(k)494 2655 y Fo(x)522 2635 y Fk(k)543 2570 y Fl(!)582 2581 y Fr(\000)p Fk(m)684 2655 y Fp(=)42 b(1)11 b(+)867 2593 y Fr(1)848 2608 y Fl(X)852 2714 y Fk(k)q Fs(=1)928 2655 y Fo(\033)958 2635 y Fk(m)956 2668 y(k)991 2655 y Fp(\()p Fo(a)1036 2662 y Fs(1)1056 2655 y Fo(;)d(a)1104 2662 y Fs(2)1123 2655 y Fo(;)g(:)g(:)g(:)15 b(;)8 b(a)1266 2662 y Fk(k)1287 2655 y Fp(\))g Fo(x)1342 2635 y Fk(k)1412 2655 y Fp(for)17 b Fo(m)c Fn(\025)h Fp(1)p Fo(:)926 2819 y Fp(3)p eop %%Page: 4 4 4 3 bop 202 286 a Fp(Let)16 b Fn(f)p Fo(\014)342 293 y Fk(n)365 286 y Fn(g)390 268 y Fr(1)390 299 y Fk(n)p Fs(=0)475 286 y Fp(b)q(e)g(a)h(sequence)e(of)i(real)e(n)o(um)o(b)q(ers) g(de\014ned)h(recursiv)o(ely)l(,)408 378 y Fo(\014)436 385 y Fs(0)469 378 y Fp(=)e(1)p Fo(;)359 495 y(\014)387 502 y Fk(n)p Fs(+1)469 495 y Fp(=)521 424 y Fl(\022)557 495 y Fo(n)d Fn(\000)652 461 y Fp(3)p 652 483 25 2 v 652 529 a(4)682 424 y Fl(\023)727 495 y Fo(\014)755 502 y Fk(n)789 495 y Fp(+)905 432 y Fk(n)880 447 y Fl(X)872 553 y Fk(j;k)q Fs(=0)838 587 y Fk(j)r Fs(+)p Fk(k)q Fs(=)p Fk(n)p Fs(+1)1003 495 y Fo(\014)1031 502 y Fk(j)1049 495 y Fo(\014)1077 502 y Fk(k)1108 495 y Fn(\000)1247 432 y Fk(n)1222 447 y Fl(X)1201 553 y Fk(j;k)q(;`)p Fs(=0)1158 587 y Fk(j)r Fs(+)p Fk(k)q Fs(+)p Fk(`)p Fs(=)p Fk(n)p Fs(+1)1365 495 y Fo(\014)1393 502 y Fk(j)1411 495 y Fo(\014)1439 502 y Fk(k)1460 495 y Fo(\014)1488 502 y Fk(`)1512 495 y Fo(:)1685 480 y Fp(\(6\))129 671 y(Here)k(are)h(sev)o(eral)f(\014rst) i(v)m(alues:)369 774 y Fo(\014)397 781 y Fs(1)430 774 y Fp(=)d Fn(\000)526 740 y Fp(3)p 526 763 V 526 808 a(4)555 774 y Fo(;)24 b(\014)621 781 y Fs(2)654 774 y Fp(=)14 b Fn(\000)750 740 y Fp(21)p 750 763 49 2 v 750 808 a(16)803 774 y Fo(;)25 b(\014)870 781 y Fs(3)903 774 y Fp(=)13 b Fn(\000)998 740 y Fp(165)p 998 763 74 2 v 1010 808 a(32)1076 774 y Fo(;)24 b(\014)1142 781 y Fs(4)1175 774 y Fp(=)14 b Fn(\000)1271 740 y Fp(7245)p 1271 763 98 2 v 1283 808 a(256)1373 774 y Fo(;)33 b(:)8 b(:)g(:)16 b(:)202 880 y Fp(F)l(or)25 b(a)g(\014xed)f(constan)o(t)h Fo(c)k Fn(2)f Fm(R)22 b Fp(w)o(e)i(in)o(tro)q(duce)g(a)h(sequence)f(of) h(p)q(olynomials,)129 941 y Fo(p)153 948 y Fk(n)177 941 y Fp(\()p Fo(c)p Fp(;)8 b Fo(z)r Fp(\))13 b Fn(2)h Fm(R)p Fp([)p Fo(z)q Fp(],)f Fo(n)h Fn(2)g Fm(Z)587 948 y Fs(+)614 941 y Fp(,)h(b)o(y)h(the)g(recursiv)o(e)f(rule)765 1034 y Fo(p)789 1041 y Fs(0)809 1034 y Fp(\()p Fo(c)p Fp(;)8 b Fo(z)r Fp(\))14 b(=)f(3)p Fo(z)h Fn(\000)c Fo(c)574 b Fp(\(7\))129 1128 y(and)209 1254 y Fo(p)233 1261 y Fk(n)298 1254 y Fp(=)42 b(3)8 b Fo(\033)440 1233 y Fs(0)438 1266 y Fk(n)462 1254 y Fp(\()p Fo(p)505 1261 y Fs(0)525 1254 y Fo(;)g(p)571 1261 y Fs(1)591 1254 y Fo(;)g(:)g(:)g(:)16 b(;)8 b(p)733 1261 y Fk(n)p Fr(\000)p Fs(1)802 1254 y Fp(\))j(+)883 1191 y Fk(n)p Fr(\000)p Fs(1)881 1206 y Fl(X)884 1312 y Fk(k)q Fs(=1)966 1220 y Fp(4)990 1202 y Fk(k)q Fs(+1)1057 1220 y Fo(\014)1085 1227 y Fk(k)q Fs(+1)p 966 1242 185 2 v 1045 1288 a Fo(k)1164 1254 y(\033)1194 1233 y Fk(k)1192 1266 y(n)p Fr(\000)p Fk(k)1262 1254 y Fp(\()p Fo(p)1305 1261 y Fs(0)1325 1254 y Fo(;)d(p)1371 1261 y Fs(1)1391 1254 y Fo(;)g(:)g(:)g(:)16 b(;)8 b(p)1533 1261 y Fk(n)p Fr(\000)p Fk(k)q Fr(\000)p Fs(1)1648 1254 y Fp(\))378 1407 y(+)421 1373 y(4)445 1355 y Fk(n)p Fs(+1)514 1373 y Fo(\014)542 1380 y Fk(n)p Fs(+1)p 421 1396 190 2 v 501 1441 a Fo(n)623 1407 y(:)1048 b Fp(\(8\))129 1513 y(This)16 b(can)g(b)q(e)h(rewritten)e(with)h(the)g(aid)h(of)f(the) g(p)q(olynomials)f Fo(s)1317 1520 y Fk(m;k)1380 1513 y Fp(,)254 1646 y Fo(p)278 1653 y Fk(n)344 1646 y Fp(=)41 b(3)481 1584 y Fk(n)456 1599 y Fl(X)461 1704 y Fk(j)r Fs(=1)541 1613 y Fp(\()p Fn(\000)p Fp(1\))642 1595 y Fk(j)r Fs(+1)p 541 1635 165 2 v 612 1680 a Fo(j)719 1646 y(s)742 1653 y Fk(j;n)789 1646 y Fp(\()p Fo(p)832 1653 y Fs(0)852 1646 y Fo(;)8 b(p)898 1653 y Fs(1)918 1646 y Fo(;)g(:)g(:)g(:)16 b(;)8 b(p)1060 1653 y Fk(n)p Fr(\000)p Fs(1)1129 1646 y Fp(\))423 1822 y(+)480 1760 y Fk(n)p Fr(\000)p Fs(1)478 1775 y Fl(X)481 1881 y Fk(k)q Fs(=0)563 1789 y Fp(4)587 1771 y Fk(n)p Fr(\000)p Fk(k)q Fs(+1)702 1789 y Fo(\014)730 1796 y Fk(n)p Fr(\000)p Fk(k)q Fs(+1)p 563 1811 283 2 v 645 1856 a Fo(n)k Fn(\000)e Fo(k)885 1760 y Fk(k)859 1775 y Fl(X)864 1880 y Fk(j)r Fs(=0)939 1752 y Fl(\022)975 1789 y Fn(\000)p Fo(n)h Fp(+)20 b Fo(k)1046 1856 y(j)1139 1752 y Fl(\023)1184 1822 y Fo(s)1207 1829 y Fk(j;k)1252 1822 y Fp(\()p Fo(p)1295 1829 y Fs(0)1315 1822 y Fo(;)8 b(p)1361 1829 y Fs(1)1381 1822 y Fo(;)g(:)g(:)g(:)16 b(;)8 b(p)1523 1829 y Fk(k)q Fr(\000)p Fs(1)1590 1822 y Fp(\))p Fo(:)129 1966 y Fp(F)l(or)21 b Fo(n)j Fn(\025)e Fp(1,)h(the)f(degree)f(of)h Fo(p)728 1973 y Fk(n)752 1966 y Fp(\()p Fo(c)p Fp(;)8 b Fo(z)r Fp(\))21 b(is)g(less)h(or)g (equal)f(to)h Fo(n)f Fp(\(this)h(can)g(b)q(e)f(easily)129 2026 y(sho)o(wn)15 b(b)o(y)f(induction)g(when)h(using)f(the)h(fact)f (that)h(for)g(an)o(y)f(monomial)f Fo(a)1507 2005 y Fk(s)1523 2010 y Fj(1)1501 2039 y Fk(i)1513 2044 y Fj(1)1542 2026 y Fo(a)1574 2005 y Fk(s)1590 2010 y Fj(2)1568 2039 y Fk(i)1580 2044 y Fj(2)1617 2026 y Fo(:)8 b(:)g(:)g(a)1715 2003 y Fk(s)1731 2009 y Fi(`)1709 2039 y Fk(i)1721 2045 y Fi(`)129 2087 y Fp(o)q(ccurring)20 b(in)f Fo(\033)438 2069 y Fk(m)436 2100 y(k)471 2087 y Fp(\()p Fo(a)516 2094 y Fs(1)535 2087 y Fo(;)8 b(a)583 2094 y Fs(2)602 2087 y Fo(;)g(:)g(:)g(:)16 b(;)8 b(a)746 2094 y Fk(k)767 2087 y Fp(\))20 b(it)f(holds)989 2049 y Fl(P)1050 2087 y Fo(i)1067 2094 y Fk(j)1085 2087 y Fo(s)1108 2094 y Fk(j)1146 2087 y Fp(=)h Fo(k)r Fp(\).)32 b(Here)19 b(are)h(sev)o(eral)e (\014rst)129 2147 y(p)q(olynomials)d Fo(p)424 2154 y Fk(n)448 2147 y Fp(\()p Fo(z)r Fp(\),)285 2240 y Fo(p)309 2247 y Fs(1)330 2240 y Fp(\()p Fo(c)p Fp(;)8 b Fo(z)r Fp(\))41 b(=)g(9)8 b Fo(z)14 b Fn(\000)d Fp(21)g Fn(\000)g Fp(3)d Fo(c;)285 2335 y(p)309 2342 y Fs(2)330 2335 y Fp(\()p Fo(c)p Fp(;)g Fo(z)r Fp(\))41 b(=)g Fn(\000)600 2301 y Fp(27)p 600 2324 49 2 v 612 2369 a(2)662 2335 y Fo(z)687 2315 y Fs(2)718 2335 y Fp(+)11 b(\(90)g(+)g(9)p Fo(c)p Fp(\))d Fo(z)14 b Fn(\000)d Fp(228)h Fn(\000)f Fp(30)d Fo(c)k Fn(\000)1331 2301 y Fp(3)p 1331 2324 25 2 v 1331 2369 a(2)1368 2335 y Fo(c)1389 2315 y Fs(2)1409 2335 y Fo(;)285 2458 y(p)309 2465 y Fs(3)330 2458 y Fp(\()p Fo(c)p Fp(;)c Fo(z)r Fp(\))41 b(=)g(27)8 b Fo(z)637 2438 y Fs(3)669 2458 y Fn(\000)719 2388 y Fl(\022)760 2424 y Fp(621)p 760 2447 74 2 v 785 2492 a(2)850 2458 y(+)j(27)d Fo(c)976 2388 y Fl(\023)1022 2458 y Fo(z)1047 2438 y Fs(2)1077 2458 y Fp(+)j(\(1638)i(+)e(207)d Fo(c)k Fp(+)f(9)d Fo(c)1518 2438 y Fs(2)1539 2458 y Fp(\))g Fo(z)556 2588 y Fn(\000)p Fp(3540)13 b Fn(\000)d Fp(546)e Fo(c)13 b Fn(\000)922 2554 y Fp(69)p 922 2576 49 2 v 934 2622 a(2)984 2588 y Fo(c)1005 2567 y Fs(2)1035 2588 y Fn(\000)e Fo(c)1106 2567 y Fs(3)1126 2588 y Fo(:)202 2694 y Fp(No)o(w)16 b(w)o(e)g(are)g(able)g(to)h(form)o(ulate)d(the)i(result.)926 2819 y(4)p eop %%Page: 5 5 5 4 bop 129 286 a Fh(Theorem)16 b(1.)24 b Fg(F)l(or)f(any)h(initial)g (data)f Fp(\()p Fo(t)931 293 y Fs(0)950 286 y Fo(;)8 b(h)1000 293 y Fs(0)1020 286 y Fo(;)g(h)1070 293 y Fs(1)1090 286 y Fp(\))25 b Fn(2)g Fm(R)p Fn(\002)7 b Fp(]0)p Fo(;)h Fn(1)p Fp([)g Fn(\002)p Fm(R)17 b Fg(ther)n(e)24 b(exists)129 347 y(a)d(unique)i(solution)g Fo(h)p Fp(\()p Fo(t)p Fp(\))e Fg(to)h(the)g(pr)n(oblem)f(\(1)p (#equation.1) [[308 634 314 646] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)i (\(2)p (#equation.2) [[333 634 339 646] [1 1 1 [3 3]] [0 0 1]] pdfm (\))e(on)h(the)g(r)n(e)n(al)f(line.)36 b(Mor)n(e)n(over,)129 407 y(ther)n(e)16 b(exists)i(a)f(c)n(onstant)g Fo(c)d Fp(=)g Fo(c)p Fp(\()p Fo(t)762 414 y Fs(0)781 407 y Fo(;)8 b(h)831 414 y Fs(0)851 407 y Fo(;)g(h)901 414 y Fs(1)920 407 y Fp(\))14 b Fn(2)g Fm(R)g Fg(such)j(that)g(it)g(holds,)g(for)f(al) r(l)i Fo(n)c Fn(2)g Fm(Z)1721 414 y Fs(+)129 467 y Fg(and)j Fo(t)d Fn(!)f Fp(+)p Fn(1)p Fg(:)327 590 y Fo(h)p Fp(\()p Fo(t)p Fp(\))h(=)g(\(4)p Fo(t)p Fp(\))557 569 y Fs(1)p Fk(=)p Fs(4)620 504 y Fl( )659 590 y Fp(1)e(+)769 527 y Fk(n)744 542 y Fl(X)748 648 y Fk(k)q Fs(=1)829 556 y Fo(q)851 563 y Fk(k)872 556 y Fp(\()p Fo(c)p Fp(;)c(ln)o(\(4)p Fo(t)p Fp(\)\))p 829 578 245 2 v 932 624 a Fo(t)950 609 y Fk(k)1089 590 y Fp(+)j Fo(O)1176 534 y Fl(\020)1215 519 y(\022)1257 556 y Fp(ln)o(\()p Fo(t)p Fp(\))p 1257 578 97 2 v 1296 624 a Fo(t)1358 519 y Fl(\023)1395 531 y Fk(n)p Fs(+1)1471 534 y Fl(\021)1501 504 y(!)1685 590 y Fp(\(9\))129 720 y Fg(wher)n(e)520 842 y Fo(q)542 849 y Fk(k)576 842 y Fp(=)657 779 y Fk(k)630 794 y Fl(X)628 899 y Fk(m)p Fs(=1)728 808 y Fp(1)p 718 830 46 2 v 718 876 a(4)742 861 y Fk(k)777 771 y Fl(\022)826 788 y Fs(1)p 826 796 18 2 v 826 825 a(4)813 876 y Fo(m)856 771 y Fl(\023)893 842 y Fo(s)916 849 y Fk(m;k)978 842 y Fp(\()p Fo(p)1021 849 y Fs(0)1041 842 y Fo(;)d(p)1087 849 y Fs(1)1107 842 y Fo(;)g(:)g(:)g(:)16 b(;)8 b(p)1249 849 y Fk(k)q Fr(\000)p Fs(1)1316 842 y Fp(\))g Fo(:)129 973 y Fg(The)17 b(de)n(gr)n(e)n(e)g (of)h Fo(q)453 980 y Fk(k)474 973 y Fp(\()p Fo(c)p Fp(;)8 b Fo(z)r Fp(\))17 b Fg(is)g(less)h(than)g(or)f(e)n(qual)h(to)g Fo(k)r Fg(.)129 1055 y(R)n(emarks.)23 b Fp(\(i\))16 b(Sev)o(eral)f (\014rst)h(p)q(olynomials)g Fo(q)981 1062 y Fk(k)1002 1055 y Fp(\()p Fo(c)p Fp(;)8 b Fo(z)r Fp(\))16 b(are)151 1165 y Fo(q)173 1172 y Fs(1)192 1165 y Fp(\()p Fo(c)p Fp(;)8 b Fo(z)r Fp(\))41 b(=)436 1132 y(3)p 424 1154 49 2 v 424 1199 a(16)486 1165 y Fo(z)13 b Fn(\000)589 1132 y Fp(1)p 577 1154 V 577 1199 a(16)639 1165 y Fo(c;)151 1288 y(q)173 1295 y Fs(2)192 1288 y Fp(\()p Fo(c)p Fp(;)8 b Fo(z)r Fp(\))41 b(=)h Fn(\000)475 1255 y Fp(27)p 463 1277 74 2 v 463 1322 a(512)549 1288 y Fo(z)574 1268 y Fs(2)605 1288 y Fp(+)654 1218 y Fl(\022)707 1255 y Fp(9)p 695 1277 49 2 v 695 1322 a(64)760 1288 y(+)838 1255 y(9)p 814 1277 74 2 v 814 1322 a(256)900 1288 y Fo(c)921 1218 y Fl(\023)966 1288 y Fo(z)13 b Fn(\000)1057 1255 y Fp(21)p 1057 1277 49 2 v 1057 1322 a(64)1122 1288 y Fn(\000)1189 1255 y Fp(3)p 1177 1277 V 1177 1322 a(64)1239 1288 y Fo(c)e Fn(\000)1350 1255 y Fp(3)p 1326 1277 74 2 v 1326 1322 a(512)1412 1288 y Fo(c)1433 1268 y Fs(2)1453 1288 y Fo(;)151 1424 y(q)173 1431 y Fs(3)192 1424 y Fp(\()p Fo(c)p Fp(;)d Fo(z)r Fp(\))41 b(=)436 1391 y(189)p 424 1413 98 2 v 424 1459 a(8192)526 1424 y Fo(z)551 1404 y Fs(3)582 1424 y Fn(\000)632 1354 y Fl(\022)686 1391 y Fp(135)p 673 1413 V 673 1459 a(1024)787 1424 y(+)853 1391 y(189)p 841 1413 V 841 1459 a(8192)952 1424 y Fo(c)973 1354 y Fl(\023)1018 1424 y Fo(z)1043 1404 y Fs(2)1073 1424 y Fp(+)1122 1354 y Fl(\022)1176 1391 y Fp(549)p 1164 1413 V 1164 1459 a(1024)1278 1424 y(+)1344 1391 y(45)p 1332 1413 74 2 v 1332 1459 a(512)1418 1424 y Fo(c)11 b Fp(+)1529 1391 y(63)p 1504 1413 98 2 v 1504 1459 a(8192)1615 1424 y Fo(c)1636 1404 y Fs(2)1656 1354 y Fl(\023)1701 1424 y Fo(z)419 1554 y Fn(\000)463 1520 y Fp(57)p 463 1543 49 2 v 463 1588 a(64)527 1554 y Fn(\000)594 1520 y Fp(183)p 582 1543 98 2 v 582 1588 a(1024)693 1554 y Fo(c)g Fn(\000)804 1520 y Fp(15)p 780 1543 V 780 1588 a(1024)891 1554 y Fo(c)912 1534 y Fs(2)942 1554 y Fn(\000)1034 1520 y Fp(7)p 997 1543 V 997 1588 a(8192)1108 1554 y Fo(c)1129 1534 y Fs(3)1149 1554 y Fo(:)202 1657 y Fp(\(ii\))19 b(In)h(the)g(\014nal)h(step)f(of)h(the)f(pro)q(of,)i(in)e(Subsection)g (4.5)p (#subsection.4.5) [[375 319 390 331] [1 1 1 [3 3]] [0 0 1]] pdfm (,)i (w)o(e)e(shall)g(sho)o(w)h(the)129 1717 y(follo)o(wing)16 b(in)o(v)m(ariance)f(prop)q(ert)o(y)h(of)h(the)f(asymptotic)f (expansion.)21 b(Set)495 1846 y Fo(A)532 1853 y Fk(n)555 1846 y Fp(\()p Fo(c)p Fp(;)8 b Fo(t)p Fp(\))13 b(=)h(\(4)p Fo(t)p Fp(\))799 1825 y Fs(1)p Fk(=)p Fs(4)862 1761 y Fl( )901 1846 y Fp(1)e(+)1011 1784 y Fk(n)986 1799 y Fl(X)990 1905 y Fk(k)q Fs(=1)1071 1812 y Fo(q)1093 1819 y Fk(k)1114 1812 y Fp(\()p Fo(c)p Fp(;)c(ln)o(\(4)p Fo(t)p Fp(\)\))p 1071 1835 245 2 v 1174 1880 a Fo(t)1192 1866 y Fk(k)1320 1761 y Fl(!)1368 1846 y Fo(;)129 1976 y Fp(with)16 b Fo(n)e Fn(2)g Fm(Z)366 1983 y Fs(+)409 1976 y Fp(and)j Fo(c;)8 b(t)13 b Fn(2)h Fm(R)p Fp(.)k(Then)f(for)f(all)g Fo(s)e Fn(2)g Fm(R)f Fp(it)j(holds)g(true)g(that)247 2099 y Fo(A)284 2106 y Fk(n)307 2099 y Fp(\()p Fo(c)p Fp(;)8 b Fo(t)i Fp(+)h Fo(s)p Fp(\))j(=)g Fo(A)591 2106 y Fk(n)614 2099 y Fp(\()p Fo(c)d Fn(\000)g Fp(4)p Fo(s)p Fp(;)d Fo(t)p Fp(\))j(+)g Fo(t)899 2078 y Fs(1)p Fk(=)p Fs(4)962 2099 y Fo(O)1000 2044 y Fl(\020)1038 2029 y(\022)1080 2065 y Fp(ln\()p Fo(t)p Fp(\))p 1080 2088 97 2 v 1119 2133 a Fo(t)1181 2029 y Fl(\023)1218 2040 y Fk(n)p Fs(+1)1295 2044 y Fl(\021)1373 2099 y Fp(as)17 b Fo(t)d Fn(!)f Fp(+)p Fn(1)p Fo(:)202 2213 y Fp(The)18 b(remainder)f(of)i(the)f(pap)q(er)h (con)o(tains)g(all)f(necessary)g(steps)h(to)g(pro)o(v)o(e)f(Theo-)129 2273 y(rem)d(1)p (#thm.1) [[126 171 132 183] [1 1 1 [3 3]] [0 0 1]] pdfm (.)25 b(W)l(e)17 b(shall)g(pro)q(ceed)h(as)g(follo)o(ws.)24 b(In)17 b(Section)g(2)p (#section.2) [[348 171 354 183] [1 1 1 [3 3]] [0 0 1]] pdfm 18 w(w)o(e)g(sho)o(w)h(the)f(completeness)129 2333 y(and)22 b(deriv)o(e)e(the)i(\014rst)g(term)e(of)i(the)g(asymptotic)f(series.)37 b(In)22 b(Section)f(3)p (#section.3) [[435 157 441 169] [1 1 1 [3 3]] [0 0 1]] pdfm 22 w(w)o(e)h(mak)o(e)129 2393 y(use)16 b(of)h(the)f(fact)g(that)h(the)f (second)g(order)h(di\013eren)o(tial)e(equation)h(can)g(b)q(e)h(reduced) e(to)129 2453 y(a)21 b(\014rst)h(order)f(di\013eren)o(tial)f(equation)h (and)h(w)o(e)e(in)o(v)o(estigate)g(the)h(asymptotic)f(prop-)129 2513 y(erties)f(of)h(the)f(latter)h(equation.)32 b(These)19 b(results)h(are)g(used)g(in)f(Section)h(4)p (#section.4) [[435 114 441 126] [1 1 1 [3 3]] [0 0 1]] pdfm 20 w(to)g(deriv)o(e)129 2574 y(the)f(asymptotic)f(prop)q(erties)h(of)h (the)f(original)g(second)g(order)h(di\013eren)o(tial)e(equation)129 2634 y(and)j(this)f(w)o(a)o(y)h(w)o(e)f(complete)e(the)j(pro)q(of)g(of) g(Theorem)f(1)p (#thm.1) [[362 85 368 97] [1 1 1 [3 3]] [0 0 1]] pdfm (.)34 b(Section)20 b(5)p (#section.5) [[421 85 427 97] [1 1 1 [3 3]] [0 0 1]] pdfm 21 w(con)o(tains)h(an)129 2694 y(additional)16 b(remark)f(on)h(the)g (asymptotics)g(of)g(the)g(Lam)o(b)q(ert)g(function.)926 2819 y(5)p eop %%Page: 6 6 6 5 bop 129 286 a Fq(2)81 b(Basic)18 b(prop)r(erties)h(of)f(the)h (di\013eren)n(tial)e(equation)129 396 y Fp(The)f(di\013eren)o(tial)f (equation)h(\(1)p (#equation.1) [[238 622 244 634] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h(is) f(equiv)m(alen)o(t)e(to)j(the)f(dynamical)e(system)472 530 y(\()p Fo(x)519 510 y Fr(0)531 530 y Fo(;)8 b(y)579 510 y Fr(0)590 530 y Fp(\))13 b(=)674 460 y Fl(\022)711 530 y Fo(y)r(;)775 496 y Fp(1)p 763 519 48 2 v 763 564 a Fo(x)791 550 y Fs(3)827 530 y Fn(\000)e Fo(y)903 460 y Fl(\023)996 530 y Fp(on)17 b Fo(M)i Fp(=)d(]0)p Fo(;)8 b Fn(1)p Fp([)g Fn(\002)p Fm(R)p Fo(:)253 b Fp(\(10\))129 702 y Fh(Prop)r(osition)17 b(2.)24 b Fg(The)i(\015ow)g(of)f(\(10)p (#equation.10) [[265 549 277 561] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (is)h(c)n(omplete)g(and)g(so)g(for)e(al)r(l)j(initial)g(data)129 762 y Fp(\()p Fo(t)166 769 y Fs(0)185 762 y Fo(;)8 b(h)235 769 y Fs(0)255 762 y Fo(;)g(h)305 769 y Fs(1)324 762 y Fp(\))15 b Fn(2)h Fm(R)p Fn(\002)7 b Fp(]0)p Fo(;)h Fn(1)p Fp([)g Fn(\002)p Fm(R)k Fg(ther)n(e)18 b(exists)h(a)f(unique)h (glob)n(al)r(ly)h(de\014ne)n(d)f(p)n(ositive)f(so-)129 822 y(lution)g Fo(h)g Fg(of)f(\(1)p (#equation.1) [[166 520 172 532] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (with)h(initial)g(c)n(onditions)g Fo(h)p Fp(\()p Fo(t)1000 829 y Fs(0)1019 822 y Fp(\))c(=)g Fo(h)1132 829 y Fs(0)1152 822 y Fg(,)j Fo(h)1212 804 y Fr(0)1224 822 y Fp(\()p Fo(t)1261 829 y Fs(0)1280 822 y Fp(\))d(=)g Fo(h)1393 829 y Fs(1)1413 822 y Fg(.)129 920 y(Pr)n(o)n(of.)22 b Fp(W)l(e)f(use)g(the)f(follo)o(wing)g(criterion)g(\(c.f.)34 b([1)p (#cite.Abraham-Marsden) [[333 496 339 508] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(,)22 b(c)o(hap.)34 b(2.1.20]\):)d Fg(The)21 b(\015ow)h(of)129 980 y(a)f Fo(C)214 962 y Fs(1)256 980 y Fg(ve)n(ctor)h(\014eld)h Fo(\030)h Fg(on)f(a)e(manifold)i Fo(M)k Fg(is)22 b(c)n(omplete)h(if)e (ther)n(e)h(is)g(a)g(pr)n(op)n(er)e(map)129 1041 y Fo(f)f Fn(2)14 b Fo(C)258 1023 y Fs(1)277 1041 y Fp(\()p Fo(M)r(;)8 b Fm(R)p Fp(\))15 b Fg(which)j(me)n(ets)g(the)g(estimate)392 1146 y Fn(9)p Fo(A)13 b(>)g Fp(0)p Fo(;)8 b(B)17 b(>)d Fp(0)p Fo(;)i Fn(8)p Fo(p)d Fn(2)h Fo(M)r(;)59 b Fn(j)p Fo(\030)13 b Fn(\001)e Fo(f)5 b Fp(\()p Fo(p)p Fp(\))p Fn(j)15 b(\024)e Fo(A)8 b Fn(j)p Fo(f)d Fp(\()p Fo(p)p Fp(\))p Fn(j)12 b Fp(+)f Fo(B)s(:)144 1252 y Fp(In)k(our)g(case)h Fo(M)j Fp(=)d(]0)p Fo(;)8 b Fn(1)p Fp([)g Fn(\002)p Fm(R)p Fp(,)k Fo(\030)k Fp(=)e Fo(y)r(@)889 1259 y Fk(x)919 1252 y Fp(+)9 b(\()p Fo(x)1013 1234 y Fr(\000)p Fs(3)1071 1252 y Fn(\000)i Fo(y)r Fp(\))d Fo(@)1200 1259 y Fk(y)1235 1252 y Fp(and)16 b(w)o(e)f(c)o(ho)q(ose)g Fo(f)5 b Fp(\()p Fo(x;)j(y)r Fp(\))14 b(=)129 1312 y Fo(x)157 1294 y Fs(2)187 1312 y Fp(+)d Fo(y)262 1294 y Fs(2)293 1312 y Fp(+)g Fo(x)370 1294 y Fr(\000)p Fs(2)416 1312 y Fp(.)22 b(With)16 b(this)g(c)o(hoice)f(w)o(e)h(ha)o(v)o(e)423 1417 y Fn(j)p Fo(\030)e Fn(\001)d Fo(f)5 b Fp(\()p Fo(x;)j(y)r Fp(\))p Fn(j)13 b Fp(=)h Fn(j)p Fp(2)p Fo(xy)e Fn(\000)f Fp(2)p Fo(y)921 1397 y Fs(2)941 1417 y Fn(j)i(\024)h Fo(x)1049 1397 y Fs(2)1079 1417 y Fp(+)d(3)p Fo(y)1178 1397 y Fs(2)1212 1417 y Fn(\024)i Fp(3)8 b Fo(f)d Fp(\()p Fo(x;)j(y)r Fp(\))p Fo(:)129 1523 y Fp(Moreo)o(v)o(er,)16 b(for)h(an)o(y)g(b)q (ounded)i(set)e Fo(S)h Fn(\032)e Fm(R)e Fp(the)j(in)o(v)o(erse)f(image) g Fo(f)1379 1505 y Fr(\000)p Fs(1)1426 1523 y Fp(\()p Fo(S)s Fp(\))i(is)f(b)q(ounded)129 1583 y(and)g(separated)h(from)e(the) h(b)q(order)g(of)h(the)f(half-plane)g Fo(M)5 b Fp(:)23 b(there)17 b(exists)f Fo(")f(>)g Fp(0)j(suc)o(h)129 1643 y(that)g Fo(f)265 1625 y Fr(\000)p Fs(1)312 1643 y Fp(\()p Fo(S)s Fp(\))e Fn(\032)g Fp([)8 b Fo(";)p Fn(1)g Fp(])j Fn(\002)g Fm(R)p Fp(.)22 b(This)c(implies)d(that)j Fo(f)k Fp(is)c(in)f(fact)g(a)h(prop)q(er)g(map)f(and)129 1704 y(the)f(prop)q(osition)h(is)f(pro)o(v)o(en.)p 1712 1704 2 33 v 1714 1672 30 2 v 1714 1704 V 1743 1704 2 33 v 129 1838 a Fh(Prop)r(osition)h(3.)24 b Fg(L)n(et)d Fp(\()p Fo(x)p Fp(\()p Fo(t)p Fp(\))p Fo(;)8 b(y)r Fp(\()p Fo(t)p Fp(\)\))p Fg(,)20 b(with)i Fo(t)e Fn(2)g Fp([)8 b Fo(t)1089 1845 y Fs(0)1109 1838 y Fo(;)g Fn(1)p Fp([)g Fg(,)21 b(b)n(e)g(a)g(solution)h(of)f(the)h(dy-)129 1898 y(namic)n(al)c(system) f(\(10)p (#equation.10) [[190 261 202 273] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)22 b(Then)c(ther)n(e)f(exists)i Fo(T)h Fn(2)14 b Fp([)8 b Fo(t)1115 1905 y Fs(0)1134 1898 y Fo(;)g Fn(1)p Fp([)25 b Fg(such)18 b(that)231 2003 y Fn(8)p Fo(t;)8 b(s;)14 b(t)g Fn(\025)f Fo(s)h Fn(\025)g Fo(T)t(;)629 1960 y Fn(p)p 671 1960 25 2 v 43 x Fp(2)8 b(\()p Fo(t)j Fp(+)g Fo(c)p Fp(\()p Fo(s)p Fp(\)\))901 1983 y Fs(1)p Fk(=)p Fs(4)970 2003 y Fn(\024)i Fo(x)p Fp(\()p Fo(t)p Fp(\))g Fn(\024)h Fo(y)r Fp(\()p Fo(s)p Fp(\))c(+)1318 1960 y Fn(p)p 1360 1960 V 43 x Fp(2)f(\()p Fo(t)h Fp(+)h Fo(c)p Fp(\()p Fo(s)p Fp(\)\))1590 1983 y Fs(1)p Fk(=)p Fs(4)129 2109 y Fg(wher)n(e)17 b Fo(c)p Fp(\()p Fo(s)p Fp(\))d(=)419 2089 y Fs(1)p 419 2097 18 2 v 419 2126 a(4)450 2109 y Fo(x)p Fp(\()p Fo(s)p Fp(\))539 2091 y Fs(4)569 2109 y Fn(\000)d Fo(s)p Fg(.)129 2207 y(Pr)n(o)n(of.)22 b Fp(Set)16 b(\(in)g(this)g(pro)q(of)s(\))i Fo(g)r Fp(\()p Fo(x;)8 b(y)r Fp(\))13 b(=)h Fo(x)918 2189 y Fr(\000)p Fs(3)976 2207 y Fn(\000)d Fo(y)18 b Fp(and)350 2342 y Fo(g)373 2349 y Fs(1)393 2342 y Fp(\()p Fo(x;)8 b(y)r Fp(\))13 b(=)g Fn(\000)p Fp(3)659 2308 y Fo(y)p 648 2330 48 2 v 648 2376 a(x)676 2362 y Fs(4)711 2342 y Fn(\000)778 2308 y Fp(1)p 766 2330 V 766 2376 a Fo(x)794 2362 y Fs(3)830 2342 y Fp(+)e Fo(y)k Fp(=)f Fn(\000)1017 2287 y Fl(\020)1047 2342 y Fp(1)d(+)g(3)1170 2308 y Fo(y)p 1169 2330 28 2 v 1169 2376 a(x)1201 2287 y Fl(\021)1240 2342 y Fo(g)r Fp(\()p Fo(x;)d(y)r Fp(\))i Fn(\000)h Fp(3)1476 2308 y Fo(y)1502 2290 y Fs(2)p 1476 2330 46 2 v 1485 2376 a Fo(x)129 2468 y Fp(on)16 b Fo(M)5 b Fp(.)22 b(Th)o(us)17 b Fo(x)436 2450 y Fr(0)461 2468 y Fp(=)d Fo(y)r Fp(,)h Fo(y)594 2450 y Fr(0)619 2468 y Fp(=)f Fo(g)r Fp(\()p Fo(x;)8 b(y)r Fp(\))15 b(and)931 2449 y Fk(d)p 925 2457 31 2 v 925 2485 a(dt)961 2468 y Fo(g)r Fp(\()p Fo(x)p Fp(\()p Fo(t)p Fp(\))p Fo(;)8 b(y)r Fp(\()p Fo(t)p Fp(\)\))k(=)i Fo(g)1299 2475 y Fs(1)1319 2468 y Fp(\()p Fo(x)p Fp(\()p Fo(t)p Fp(\))p Fo(;)8 b(y)r Fp(\()p Fo(t)p Fp(\)\).)202 2528 y(W)l(e)17 b(shall)f(sho)o(w)i(\014rst)f(that)g Fo(y)756 2510 y Fr(0)768 2528 y Fp(\()p Fo(t)p Fp(\))f(is)h(negativ)o (e)f(for)h(all)f(su\016cien)o(tly)f(large)i Fo(t)p Fp(.)23 b(Note)129 2589 y(that)479 2694 y Fn(8)p Fp(\()p Fo(x;)8 b(y)r Fp(\))k Fn(2)i Fo(M)r(;)57 b(g)r Fp(\()p Fo(x;)8 b(y)r Fp(\))13 b Fn(\024)h Fp(0)g(=)-8 b Fn(\))13 b Fo(g)1160 2701 y Fs(1)1180 2694 y Fp(\()p Fo(x;)8 b(y)r Fp(\))13 b Fo(<)h Fp(0)p Fo(:)926 2819 y Fp(6)p eop %%Page: 7 7 7 6 bop 129 286 a Fp(Th)o(us)18 b(if)f Fo(y)326 268 y Fr(0)337 286 y Fp(\()p Fo(s)p Fp(\))g Fn(\024)f Fp(0)i(then)g Fo(y)651 268 y Fr(0)662 286 y Fp(\()p Fo(t)p Fp(\))e Fo(<)h Fp(0)h(for)g(all)f Fo(t)f(>)h(s)p Fp(.)26 b(Hence)16 b(it)i(is)f(su\016cien)o(t)g(to)h(sho)o(w)129 347 y(that)f(there)f(is)h (at)g(least)g(one)g Fo(t)f Fp(suc)o(h)h(that)g Fo(y)952 329 y Fr(0)963 347 y Fp(\()p Fo(t)p Fp(\))d Fn(\024)h Fp(0.)23 b(Supp)q(ose)18 b(the)f(con)o(trary)l(.)23 b(Since)129 407 y(it)15 b(holds)421 517 y Fo(y)r Fp(\()p Fo(t)p Fp(\))e(=)h Fo(y)r Fp(\()p Fo(t)631 524 y Fs(0)650 517 y Fp(\))d(+)737 483 y Fo(t)g Fn(\000)g Fo(t)834 490 y Fs(0)p 734 505 123 2 v 734 551 a Fo(x)p Fp(\()p Fo(t)799 558 y Fs(0)818 551 y Fp(\))837 537 y Fs(3)872 517 y Fn(\000)922 449 y Fl(Z)972 462 y Fk(t)950 562 y(t)963 567 y Fj(0)995 447 y Fl(\022)1032 517 y Fp(3)1073 483 y Fo(t)g Fn(\000)f Fo(s)p 1069 505 109 2 v 1069 551 a(x)p Fp(\()p Fo(s)p Fp(\))1158 537 y Fs(4)1194 517 y Fp(+)h(1)1267 447 y Fl(\023)1312 517 y Fo(y)r Fp(\()p Fo(s)p Fp(\))d Fo(ds)129 639 y Fp(there)k(exists)g Fo(s)i Fn(\025)g Fo(t)489 646 y Fs(0)521 639 y Fp(suc)o(h)f(that)g Fo(y)r Fp(\()p Fo(s)p Fp(\))g Fo(>)h Fp(0.)21 b(Then)13 b(b)q(oth)g Fo(x)p Fp(\()p Fo(t)p Fp(\))g(and)g Fo(y)r Fp(\()p Fo(t)p Fp(\))f(are)h (increasing)129 699 y(p)q(ositiv)o(e)19 b(functions)h(on)g(the)g(in)o (terv)m(al)f([)8 b Fo(s;)g Fn(1)p Fp([)27 b(and,)22 b(in)d(addition,)i Fo(x)p Fp(\()p Fo(t)p Fp(\))e Fo(<)h(y)r Fp(\()p Fo(t)p Fp(\))1652 681 y Fr(\000)p Fs(1)p Fk(=)p Fs(3)1734 699 y Fp(.)129 759 y(So)c(the)g(function)g Fo(g)r Fp(\()p Fo(t)p Fp(\))e(=)g Fo(g)r Fp(\()p Fo(x)p Fp(\()p Fo(t)p Fp(\))p Fo(;)8 b(y)r Fp(\()p Fo(t)p Fp(\)\))14 b(ob)q(eys)260 880 y Fn(8)p Fo(t)e Fn(\025)i Fo(s;)57 b(g)491 859 y Fr(0)502 880 y Fp(\()p Fo(t)p Fp(\))14 b(=)f Fn(\000)670 810 y Fl(\022)707 880 y Fp(1)f(+)f(3)830 846 y Fo(y)r Fp(\()p Fo(t)p Fp(\))p 829 868 84 2 v 829 914 a Fo(x)p Fp(\()p Fo(t)p Fp(\))918 810 y Fl(\023)963 880 y Fo(g)r Fp(\()p Fo(t)p Fp(\))f Fn(\000)h Fp(3)1142 846 y Fo(y)r Fp(\()p Fo(t)p Fp(\))1224 828 y Fs(2)p 1142 868 101 2 v 1151 914 a Fo(x)p Fp(\()p Fo(t)p Fp(\))1261 880 y Fo(<)j Fn(\000)p Fp(3)8 b Fo(y)r Fp(\()p Fo(s)p Fp(\))1471 859 y Fs(7)p Fk(=)p Fs(3)1540 880 y Fo(<)14 b Fp(0)129 1000 y(whic)o(h)h(clearly)g(con)o(tradicts)h(the)g(assumption)g Fo(g)r Fp(\()p Fo(t)p Fp(\))d(=)h Fo(y)1189 982 y Fr(0)1200 1000 y Fp(\()p Fo(t)p Fp(\))g Fo(>)g Fp(0)i(for)h(all)f Fo(t)p Fp(.)202 1060 y(Let)h(no)o(w)h Fo(T)k Fn(\025)15 b Fo(t)516 1067 y Fs(0)553 1060 y Fp(b)q(e)i(suc)o(h)h(that)f Fo(y)864 1042 y Fr(0)876 1060 y Fp(\()p Fo(t)p Fp(\))e(=)g Fo(x)p Fp(\()p Fo(t)p Fp(\))1084 1042 y Fr(\000)p Fs(3)1143 1060 y Fn(\000)c Fo(y)r Fp(\()p Fo(t)p Fp(\))k Fo(<)g Fp(0)j(for)g(all)f Fo(t)e(>)g(T)24 b Fp(and)129 1120 y(\014x)15 b Fo(s)f Fn(\025)g Fo(T)7 b Fp(.)20 b(F)l(or)c(an)o(y)f Fo(t)f(>)g(T)22 b Fp(w)o(e)15 b(ha)o(v)o(e)g(\()877 1100 y Fs(1)p 877 1109 18 2 v 877 1137 a(4)899 1120 y Fo(x)p Fp(\()p Fo(t)p Fp(\))983 1102 y Fs(4)1002 1120 y Fp(\))1021 1102 y Fr(0)1047 1120 y Fp(=)f Fo(x)p Fp(\()p Fo(t)p Fp(\))1183 1102 y Fs(3)1202 1120 y Fo(y)r Fp(\()p Fo(t)p Fp(\))f Fo(>)g Fp(1.)22 b(Consequen)o(tly)l(,)14 b(if)129 1180 y Fo(t)f Fn(\025)h Fo(s)i Fp(then)595 1306 y Fo(x)p Fp(\()p Fo(t)p Fp(\))d Fn(\025)745 1262 y(p)p 786 1262 25 2 v 786 1306 a Fp(2)819 1236 y Fl(\022)855 1306 y Fo(t)e Fp(+)938 1272 y(1)p 938 1295 V 938 1340 a(4)976 1306 y Fo(x)p Fp(\()p Fo(s)p Fp(\))1065 1285 y Fs(4)1095 1306 y Fn(\000)g Fo(s)1168 1236 y Fl(\023)1205 1247 y Fs(1)p Fk(=)p Fs(4)1268 1306 y Fo(:)202 1433 y Fp(T)l(o)20 b(sho)o(w)g(the)f(other)h(inequalit)o(y)d(set,)j(for)g Fo(t)f Fn(\025)g Fo(s)p Fp(,)h Fo(z)r Fp(\()p Fo(t)p Fp(\))f(=)g Fo(x)p Fp(\()p Fo(t)p Fp(\))12 b Fn(\000)1477 1392 y(p)p 1519 1392 V 41 x Fp(2)d(\()p Fo(t)j Fp(+)i Fo(c)p Fp(\))1693 1415 y Fs(1)p Fk(=)p Fs(4)129 1494 y Fp(where)i Fo(c)d Fp(=)h Fo(c)p Fp(\()p Fo(s)p Fp(\).)21 b(W)l(e)16 b(\014nd)h(that)170 1628 y(\()p Fo(e)212 1608 y Fk(t)227 1628 y Fo(z)252 1608 y Fr(0)263 1628 y Fp(\))282 1608 y Fr(0)336 1628 y Fp(=)42 b Fo(e)439 1608 y Fk(t)453 1628 y Fp(\()p Fo(z)497 1608 y Fr(0)520 1628 y Fp(+)11 b Fo(z)594 1608 y Fr(0)o(0)615 1628 y Fp(\))22 b(=)g Fo(e)739 1608 y Fk(t)762 1543 y Fl( )818 1595 y Fp(1)p 806 1617 48 2 v 806 1663 a Fo(x)834 1648 y Fs(3)870 1628 y Fn(\000)10 b Fp(\()938 1585 y Fn(p)p 980 1585 25 2 v 43 x Fp(2)f(\()p Fo(t)h Fp(+)h Fo(c)p Fp(\))1149 1608 y Fs(1)p Fk(=)p Fs(4)1204 1628 y Fp(\))1223 1608 y Fr(\000)p Fs(3)1281 1628 y Fp(+)1335 1595 y(3)1359 1553 y Fn(p)p 1401 1553 V 42 x Fp(2)p 1335 1617 91 2 v 1356 1663 a(16)1439 1628 y(\()p Fo(t)g Fp(+)g Fo(c)p Fp(\))1576 1608 y Fr(\000)p Fs(7)p Fk(=)p Fs(4)1658 1543 y Fl(!)336 1786 y Fn(\024)421 1752 y Fp(3)445 1711 y Fn(p)p 487 1711 25 2 v 41 x Fp(2)p 421 1775 91 2 v 442 1820 a(16)524 1786 y Fo(e)547 1766 y Fk(t)570 1786 y Fp(\()p Fo(t)g Fp(+)g Fo(c)p Fp(\))707 1766 y Fr(\000)p Fs(7)p Fk(=)p Fs(4)789 1786 y Fo(:)129 1890 y Fp(It)16 b(follo)o(ws)g(that)440 2004 y Fo(z)465 1983 y Fr(0)477 2004 y Fp(\()p Fo(t)p Fp(\))d Fn(\024)h Fo(e)622 1983 y Fk(s)p Fr(\000)p Fk(t)680 2004 y Fo(z)705 1983 y Fr(0)716 2004 y Fp(\()p Fo(s)p Fp(\))d(+)842 1970 y(3)866 1929 y Fn(p)p 908 1929 25 2 v 41 x Fp(2)p 842 1992 91 2 v 863 2038 a(16)946 2004 y Fo(e)969 1983 y Fr(\000)p Fk(t)1019 1936 y Fl(Z)1069 1949 y Fk(t)1047 2049 y(s)1092 2004 y Fo(e)1115 1983 y Fk(u)1145 2004 y Fp(\()p Fo(u)g Fp(+)g Fo(c)p Fp(\))1292 1983 y Fr(\000)p Fs(7)p Fk(=)p Fs(4)1383 2004 y Fo(du)129 2118 y Fp(and)198 2232 y Fo(z)r Fp(\()p Fo(t)p Fp(\))41 b Fn(\024)g Fo(z)r Fp(\()p Fo(s)p Fp(\))11 b(+)546 2192 y Fl(\000)569 2232 y Fp(1)h Fn(\000)e Fo(e)677 2212 y Fk(s)p Fr(\000)p Fk(t)736 2192 y Fl(\001)767 2232 y Fo(z)792 2212 y Fr(0)803 2232 y Fp(\()p Fo(s)p Fp(\))h(+)929 2199 y(3)953 2157 y Fn(p)p 995 2157 25 2 v 42 x Fp(2)p 929 2221 91 2 v 950 2266 a(16)1041 2164 y Fl(Z)1091 2178 y Fk(t)1069 2277 y(s)1114 2232 y Fo(e)1137 2212 y Fr(\000)p Fk(\034)1194 2164 y Fl(Z)1244 2178 y Fk(\034)1222 2277 y(s)1274 2232 y Fo(e)1297 2212 y Fk(u)1327 2232 y Fp(\()p Fo(u)g Fp(+)g Fo(c)p Fp(\))1474 2212 y Fr(\000)p Fs(7)p Fk(=)p Fs(4)1565 2232 y Fo(du)d(d\034)320 2373 y Fn(\024)41 b Fo(z)r Fp(\()p Fo(s)p Fp(\))11 b(+)g Fo(z)571 2352 y Fr(0)583 2373 y Fp(\()p Fo(s)p Fp(\))g(+)709 2339 y(3)733 2298 y Fn(p)p 775 2298 25 2 v 41 x Fp(2)p 709 2361 91 2 v 730 2407 a(16)821 2305 y Fl(Z)870 2318 y Fk(t)848 2418 y(s)894 2332 y Fl(\000)916 2373 y Fp(1)h Fn(\000)f Fo(e)1025 2352 y Fk(u)p Fr(\000)p Fk(t)1087 2332 y Fl(\001)1126 2373 y Fp(\()p Fo(u)g Fp(+)g Fo(c)p Fp(\))1273 2352 y Fr(\000)p Fs(7)p Fk(=)p Fs(4)1364 2373 y Fo(du)320 2513 y Fn(\024)41 b Fo(z)r Fp(\()p Fo(s)p Fp(\))11 b(+)g Fo(z)571 2493 y Fr(0)583 2513 y Fp(\()p Fo(s)p Fp(\))g(+)709 2480 y(3)733 2438 y Fn(p)p 775 2438 25 2 v 42 x Fp(2)p 709 2502 91 2 v 730 2547 a(16)817 2480 y(4)p 817 2502 25 2 v 817 2547 a(3)855 2513 y(\()p Fo(s)g Fp(+)g Fo(c)p Fp(\))997 2493 y Fr(\000)p Fs(3)p Fk(=)p Fs(4)321 2603 y Fp(=)41 b Fo(z)r Fp(\()p Fo(s)p Fp(\))11 b(+)g Fo(x)574 2582 y Fr(0)586 2603 y Fp(\()p Fo(s)p Fp(\))p Fo(:)129 2694 y Fp(But)16 b Fo(z)r Fp(\()p Fo(s)p Fp(\))d(=)h(0)j(and)f(so)h Fo(x)p Fp(\()p Fo(s)p Fp(\))d Fn(\024)f Fo(y)r Fp(\()p Fo(s)p Fp(\))e(+)874 2653 y Fn(p)p 915 2653 V 915 2694 a Fp(2)e(\()p Fo(t)i Fp(+)g Fo(c)p Fp(\))1085 2676 y Fs(1)p Fk(=)p Fs(4)1140 2694 y Fp(.)p 1712 2694 2 33 v 1714 2663 30 2 v 1714 2694 V 1743 2694 2 33 v 926 2819 a(7)p eop %%Page: 8 8 8 7 bop 129 286 a Fh(Corollary)18 b(4.)24 b Fg(If)12 b Fo(h)p Fp(\()p Fo(t)p Fp(\))h Fg(is)g(a)f(solution)i(of)f(\(1)p (#equation.1) [[294 648 300 660] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g(on) g Fp([)8 b Fo(t)1090 293 y Fs(0)1109 286 y Fo(;)g Fn(1)p Fp([)21 b Fg(with)13 b(the)g(initial)h(c)n(onditions)129 347 y Fo(h)p Fp(\()p Fo(t)194 354 y Fs(0)213 347 y Fp(\))i(=)f Fo(h)329 354 y Fs(0)364 347 y Fo(>)h Fp(0)p Fg(,)j Fo(h)504 329 y Fr(0)516 347 y Fp(\()p Fo(t)553 354 y Fs(0)572 347 y Fp(\))c(=)h Fo(h)688 354 y Fs(1)708 347 y Fg(,)i(then)h(ther)n(e) g(exists)g Fo(T)j Fn(\025)15 b Fo(t)1230 354 y Fs(0)1268 347 y Fg(such)k(that)f Fo(h)1508 329 y Fr(0)1520 347 y Fp(\()p Fo(t)p Fp(\))d Fo(>)g Fp(0)k Fg(for)129 407 y(al)r(l)f Fo(t)c(>)g(T)7 b Fg(.)129 550 y Fh(Corollary)18 b(5.)24 b Fg(If)12 b Fo(h)p Fp(\()p Fo(t)p Fp(\))h Fg(is)g(a)f (solution)i(of)f(\(1)p (#equation.1) [[294 585 300 597] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g(on) g Fp([)8 b Fo(t)1090 557 y Fs(0)1109 550 y Fo(;)g Fn(1)p Fp([)21 b Fg(with)13 b(the)g(initial)h(c)n(onditions)129 610 y Fo(h)p Fp(\()p Fo(t)194 617 y Fs(0)213 610 y Fp(\))g(=)g Fo(h)326 617 y Fs(0)359 610 y Fo(>)g Fp(0)p Fg(,)k Fo(h)496 592 y Fr(0)508 610 y Fp(\()p Fo(t)545 617 y Fs(0)564 610 y Fp(\))c(=)f Fo(h)676 617 y Fs(1)696 610 y Fg(,)18 b(then)555 720 y Fo(h)p Fp(\()p Fo(t)p Fp(\))c(=)705 677 y Fn(p)p 746 677 25 2 v 746 720 a Fp(2)9 b Fo(t)797 700 y Fs(1)p Fk(=)p Fs(4)862 720 y Fp(+)i Fo(O)q Fp(\(1\))51 b Fg(as)18 b Fo(t)13 b Fn(!)h Fp(+)p Fn(1)p Fo(:)129 830 y Fg(R)n(emark.)23 b Fp(This)13 b(means)e(that)h(if)g(w)o(e)g (restrict)f(ourselv)o(es)h(in)f(what)i(follo)o(ws)f(to)h(the)f(initial) 129 890 y(condition)k(\(2)p (#equation.2) [[159 503 165 515] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (with)h Fo(h)561 897 y Fs(1)594 890 y Fo(>)d Fp(0)j(w)o(e)f(don't)g(lo) q(ose)h(the)f(generalit)o(y)f(as)i(far)f(as)h(the)f(asymp-)129 951 y(totics)h(is)g(concerned.)24 b(F)l(urthermore,)15 b(o)o(wing)j(to)g(the)f(in)o(v)m(ariance)f(of)i(the)f(di\013eren)o (tial)129 1011 y(equation)h(in)g(time)e(w)o(e)i(can)h(set)f Fo(t)764 1018 y Fs(0)801 1011 y Fp(=)g(0.)28 b(This)19 b(fact)f(will)f(b)q(e)i(used)g(in)f(the)g(course)g(of)129 1071 y(the)c(pro)q(of.)22 b(First)14 b(w)o(e)g(v)o(erify)f(Theorem)g(1) p (#thm.1) [[284 460 289 472] [1 1 1 [3 3]] [0 0 1]] pdfm 15 w(for)i(the)g(particular)f(case)g(when)h Fo(t)1545 1078 y Fs(0)1578 1071 y Fp(=)f(0)h(and)129 1131 y Fo(h)157 1138 y Fs(1)190 1131 y Fo(>)f Fp(0)i(and)g(then,)g(in)f(Subsection)g (4.5)p (#subsection.4.5) [[264 446 279 458] [1 1 1 [3 3]] [0 0 1]] pdfm (,)h (w)o(e)f(shall)h(extend)f(the)g(result)g(to)h(the)g(general)129 1191 y(initial)f(condition.)129 1358 y Fq(3)81 b(A)28 b(reduced)g(di\013eren)n(tial)f(equation)h(of)h(\014rst)f(or-)250 1449 y(der)129 1559 y Fp(In)14 b(accordance)g(with)g(the)g(remark)e(at) j(the)f(end)g(of)g(Section)g(2)p (#section.2) [[370 343 376 355] [1 1 1 [3 3]] [0 0 1]] pdfm 14 w(w)o(e)g(assume)g(that)g Fo(t)1638 1566 y Fs(0)1671 1559 y Fp(=)g(0)129 1619 y(and)j Fo(h)252 1626 y Fs(1)287 1619 y Fo(>)d Fp(0.)24 b(The)16 b(second-order)i(di\013eren)o(tial)d (equation)i(equation)g(\(1)p (#equation.1) [[420 328 426 340] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g(is) f(in)o(v)m(arian)o(t)129 1679 y(in)i Fo(t)g Fp(and)h(this)f(is)h(wh)o (y)f(it)g(can)h(b)q(e)f(reduced)g(to)h(a)g(\014rst-order)g(di\013eren)o (tial)e(equation.)129 1747 y(Actually)l(,)i(using)i(the)f(substitution) h Fo(h)p Fp(\()p Fo(t)p Fp(\))f(=)1000 1707 y Fl(\000)1023 1747 y Fo(G)1061 1729 y Fr(\000)p Fs(1)1109 1747 y Fp(\(4)p Fo(t)p Fp(\))1189 1707 y Fl(\001)1212 1718 y Fs(1)p Fk(=)p Fs(4)1267 1747 y Fp(,)h Fo(z)1325 1754 y Fs(0)1365 1747 y Fp(=)g(4)p Fo(=h)1506 1729 y Fs(4)1500 1759 y(0)1547 1747 y Fp(and)g Fo(g)1669 1754 y Fs(0)1710 1747 y Fp(=)129 1807 y Fo(h)163 1789 y Fs(3)157 1819 y(0)182 1807 y Fo(h)210 1814 y Fs(1)246 1807 y Fp(where)755 1942 y Fo(G)p Fp(\()p Fo(x)p Fp(\))14 b(=)925 1875 y Fl(Z)974 1888 y Fk(x)952 1987 y(h)972 1976 y Fj(4)972 1998 y(0)1039 1909 y Fo(ds)p 1010 1931 107 2 v 1010 1979 a(g)1043 1939 y Fl(\000)1071 1960 y Fs(4)p 1071 1968 18 2 v 1072 1996 a Fk(s)1094 1939 y Fl(\001)129 2086 y Fp(w)o(e)h(arriv)o(e)h(at)g(a)h (\014rst-order)g(nonlinear)f(di\013eren)o(tial)f(equation,)g(namely)448 2152 y Fl(\022)485 2222 y Fp(1)c Fn(\000)575 2189 y Fp(3)p 575 2211 25 2 v 575 2257 a(4)605 2222 y Fo(z)f(g)r Fp(\()p Fo(z)r Fp(\))h Fn(\000)g Fo(z)812 2202 y Fs(2)839 2222 y Fo(g)864 2202 y Fr(0)876 2222 y Fp(\()p Fo(z)r Fp(\))939 2152 y Fl(\023)984 2222 y Fo(g)r Fp(\()p Fo(z)r Fp(\))j(=)g(1)p Fo(;)24 b(g)r Fp(\()p Fo(z)1267 2229 y Fs(0)1287 2222 y Fp(\))14 b(=)f Fo(g)1394 2229 y Fs(0)1415 2222 y Fo(:)232 b Fp(\(11\))129 2361 y(W)l(e)15 b(shall)g(carry)h(out)g(the)f (computations)g(relating)g(\(1)p (#equation.1) [[341 150 347 162] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)h (\(2)p (#equation.2) [[363 150 369 162] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g(and)g(\(11)p (#equation.11) [[404 150 416 162] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (later,)f(in)g(Sub-)129 2421 y(section)g(4.1)p (#subsection.4.1) [[142 136 157 148] [1 1 1 [3 3]] [0 0 1]] pdfm (.)21 b(Here)15 b(w)o(e)g(concen)o(trate)g(on)h(the)f(study)h(of)g(\(11)p (#equation.11) [[358 136 370 148] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (on)g(the)f(in)o(terv)m(al)g([)8 b(0)p Fo(;)g(z)1693 2428 y Fs(0)1721 2421 y Fp(],)129 2481 y(assuming)16 b(that)g Fo(z)469 2488 y Fs(0)502 2481 y Fo(>)e Fp(0)j(and)g Fo(g)713 2488 y Fs(0)747 2481 y Fo(>)c Fp(0.)926 2819 y(8)p eop %%Page: 9 9 9 8 bop 129 286 a Ff(3.1)66 b(Domain)22 b(of)g(the)g(left)h(maximal)h (solution)f Fe(g)r Fz(\()p Fe(z)s Fz(\))129 379 y Fp(W)l(e)16 b(shall)g(need)g(t)o(w)o(o)g(equiv)m(alen)o(t)e(forms)i(of)g(the)g (di\013eren)o(tial)f(equation,)h(namely)599 515 y Fo(g)624 494 y Fr(0)636 515 y Fp(\()p Fo(z)r Fp(\))e(=)780 481 y(1)p 769 503 45 2 v 769 549 a Fo(z)794 534 y Fs(2)827 444 y Fl(\022)864 515 y Fp(1)d Fn(\000)986 481 y Fp(1)p 954 503 88 2 v 954 549 a Fo(g)r Fp(\()p Fo(z)r Fp(\))1047 444 y Fl(\023)1095 515 y Fn(\000)1150 481 y Fp(3)p 1150 503 25 2 v 1150 549 a(4)1184 481 y Fo(g)r Fp(\()p Fo(z)r Fp(\))p 1184 503 88 2 v 1216 549 a Fo(z)1661 515 y Fp(\(12\))129 651 y(and)592 736 y Fl(\000)614 777 y Fo(z)639 756 y Fs(3)p Fk(=)p Fs(4)694 777 y Fo(g)r Fp(\()p Fo(z)r Fp(\))782 736 y Fl(\001)805 744 y Fr(0)831 777 y Fp(=)i Fo(z)907 756 y Fr(\000)p Fs(5)p Fk(=)p Fs(4)998 707 y Fl(\022)1035 777 y Fp(1)e Fn(\000)1157 743 y Fp(1)p 1125 765 V 1125 811 a Fo(g)r Fp(\()p Fo(z)r Fp(\))1218 707 y Fl(\023)1271 777 y Fo(:)376 b Fp(\(13\))129 913 y Fg(R)n(emark.)23 b Fp(In)c(what)h(follo)o(ws)g(w)o(e)f(use)g(rep)q(eatedly)g(the)g (follo)o(wing)g(elemen)o(tary)e(argu-)129 973 y(men)o(t:)25 b Fg(if)20 b Fo( )i Fg(and)e Fo(')g Fg(ar)n(e)g(two)h(di\013er)n (entiable)g(functions)h(on)e Fp(])p Fo(a;)8 b(b)p Fp([)19 b Fg(and)h(the)h(e)n(quality)129 1033 y Fo( )r Fp(\()p Fo(z)r Fp(\))16 b(=)i Fo(')p Fp(\()p Fo(z)r Fp(\))h Fg(implies)h Fo( )615 1015 y Fr(0)626 1033 y Fp(\()p Fo(z)r Fp(\))d Fo(<)g(')793 1015 y Fr(0)805 1033 y Fp(\()p Fo(z)r Fp(\))p Fg(,)i(for)g(al)r(l)i Fo(z)e Fn(2)8 b Fp(])p Fo(a;)g(b)p Fp([)p Fg(,)18 b(then)j(the)f(two)f(functions)129 1093 y(c)n(oincide)f(in)g(at)f(most)h(one)g(p)n(oint)j Fo(z)15 b Fn(2)8 b Fp(])p Fo(a;)g(b)p Fp([.)129 1216 y Fh(Prop)r(osition)17 b(6.)24 b Fg(L)n(et)18 b Fo(g)r Fp(\()p Fo(z)r Fp(\))f Fg(b)n(e)h(the)g(left)h(maximal)f(solution)g(of)g(\(11)p (#equation.11) [[402 425 414 437] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)k (Then)c Fo(g)r Fp(\()p Fo(z)r Fp(\))g Fg(is)129 1276 y(de\014ne)n(d)g(and)g(p)n(ositive)f(on)h Fp(]0)p Fo(;)8 b(z)716 1283 y Fs(0)744 1276 y Fp(])p Fg(,)16 b(and)800 1386 y Fp(lim)808 1417 y Fk(z)q Fr(#)p Fs(0)876 1386 y Fo(g)r Fp(\()p Fo(z)r Fp(\))e(=)g(1)8 b Fo(:)585 b Fp(\(14\))129 1520 y Fg(Pr)n(o)n(of.)22 b Fp(Let)c Fo(\015)i Fp(b)q(e)d(the)g(minim)o(al)d(non-negativ)o(e)j(n)o(um)o(b)q(er)f(suc)o (h)g(that)i Fo(g)r Fp(\()p Fo(z)r Fp(\))f(is)g(de\014ned)129 1580 y(on)j(])p Fo(\015)s(;)8 b(z)287 1587 y Fs(0)314 1580 y Fp(].)30 b(Equation)20 b(\(11)p (#equation.11) [[217 338 229 350] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (clearly)d(excludes)g(the)h(p)q(ossibilit)o(y)g(that)h Fo(g)r Fp(\()p Fo(z)r Fp(\))f(=)g(0)h(for)129 1640 y(some)12 b Fo(z)k Fn(2)8 b Fp(])p Fo(\015)s(;)g(z)415 1647 y Fs(0)442 1640 y Fp(].)20 b(So)14 b Fo(g)r Fp(\()p Fo(z)r Fp(\))g(is)f(p)q (ositiv)o(e)g(on)h(])p Fo(\015)s(;)8 b(z)1033 1647 y Fs(0)1060 1640 y Fp(].)20 b(Our)13 b(goal)h(is)g(to)g(sho)o(w)g(that)g Fo(\015)i Fp(=)e(0)129 1701 y(and)j(\(14)p (#equation.14) [[130 309 142 321] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f (holds)h(true.)k(W)l(e)16 b(split)g(the)g(pro)q(of)h(in)o(to)f(six)g (claims.)202 1761 y Fg(\(i\))i Fn(9)p Fo(\030)e Fn(2)8 b Fp(])p Fo(\015)s(;)g(z)468 1768 y Fs(0)496 1761 y Fp(])15 b(s.t.)21 b Fo(g)r Fp(\()p Fo(\030)r Fp(\))15 b Fn(\025)f Fp(1.)202 1821 y(Supp)q(ose)19 b(that)g Fo(g)r Fp(\()p Fo(z)r Fp(\))f Fo(<)f Fp(1,)i Fn(8)p Fo(z)f Fn(2)8 b Fp(])p Fo(\015)s(;)g(z)919 1828 y Fs(0)947 1821 y Fp(].)27 b(Then,)19 b(b)o(y)e(\(12)p (#equation.12) [[368 280 380 292] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)j Fo(g)1360 1803 y Fr(0)1371 1821 y Fp(\()p Fo(z)r Fp(\))e Fo(<)f Fp(0)i(on)g(])p Fo(\015)s(;)8 b(z)1707 1828 y Fs(0)1734 1821 y Fp(])129 1881 y(and)17 b(so)h(there)e(exists)g(lim)613 1888 y Fk(z)q Fr(#)p Fk(\015)679 1881 y Fo(g)r Fp(\()p Fo(z)r Fp(\))f(=)f Fo(g)857 1888 y Fs(1)894 1881 y Fp(with)j Fo(g)1029 1888 y Fs(0)1064 1881 y Fo(<)e(g)1140 1888 y Fs(1)1175 1881 y Fn(\024)f Fp(1.)24 b(Hence,)15 b(b)o(y)h(minimali)o (t)o(y)129 1941 y(of)g Fo(\015)s Fp(,)g(it)g(should)h(hold)f Fo(\015)h Fp(=)d(0.)21 b(According)16 b(to)g(\(13)p (#equation.13) [[316 251 328 263] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)379 2036 y Fl(\000)402 2077 y Fo(z)427 2056 y Fs(3)p Fk(=)p Fs(4)482 2077 y Fo(g)r Fp(\()p Fo(z)r Fp(\))570 2036 y Fl(\001)592 2044 y Fr(0)618 2077 y Fo(<)e Fp(0)g(=)-8 b Fn(\))13 b Fo(g)r Fp(\()p Fo(z)r Fp(\))h Fo(>)g(g)978 2084 y Fs(0)1006 2021 y Fl(\020)1041 2043 y Fo(z)1064 2050 y Fs(0)p 1041 2065 43 2 v 1050 2111 a Fo(z)1088 2021 y Fl(\021)1118 2033 y Fs(3)p Fk(=)p Fs(4)1181 2077 y Fo(;)57 b Fn(8)p Fo(z)14 b Fn(2)8 b Fp(]0)p Fo(;)g(z)1442 2084 y Fs(0)1470 2077 y Fp(])p Fo(;)129 2202 y Fp(a)16 b(con)o(tradiction.)21 b(In)15 b(the)h(follo)o(wing)g(c)o(ho)q(ose)g Fo(\030)h Fn(2)f Fp(])p Fo(\015)s(;)8 b(z)1163 2209 y Fs(0)1191 2202 y Fp(])15 b(to)h(b)q(e)h(the)e(largest)h(n)o(um)o(b)q (er)129 2262 y(suc)o(h)g Fo(g)r Fp(\()p Fo(\030)r Fp(\))e Fn(\025)g Fp(1.)202 2323 y Fg(\(ii\))k Fo(g)r Fp(\()p Fo(z)r Fp(\))c Fo(>)f Fp(1,)j Fn(8)p Fo(z)f Fn(2)8 b Fp(])p Fo(\015)s(;)g(\030)r Fp([.)202 2383 y(Actually)l(,)14 b Fo(g)r Fp(\()p Fo(z)r Fp(\))g(=)g(1)i(implies)e Fo(g)794 2365 y Fr(0)806 2383 y Fp(\()p Fo(z)r Fp(\))f Fo(<)h Fp(0.)202 2443 y Fg(\(iii\))k Fo(\015)f Fp(=)c(0.)202 2503 y(If)18 b Fo(\015)i(>)d Fp(0)i(then,)f(b)o(y)g(\(13)p (#equation.13) [[219 116 230 128] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)h Fn(j)p Fp(\()p Fo(z)770 2485 y Fs(3)p Fk(=)p Fs(4)824 2503 y Fo(g)r Fp(\()p Fo(z)r Fp(\)\))931 2485 y Fr(0)943 2503 y Fn(j)e(\024)g Fo(\015)1058 2485 y Fr(\000)p Fs(5)p Fk(=)p Fs(4)1158 2503 y Fo(<)g Fn(1)h Fp(for)g(all)g Fo(z)h Fn(2)8 b Fp(])p Fo(\015)s(;)g(\030)r Fp([.)28 b(This)129 2563 y(means)11 b(that)i Fo(z)401 2545 y Fs(3)p Fk(=)p Fs(4)456 2563 y Fo(g)r Fp(\()p Fo(z)r Fp(\))g(is)f(absolutely)g (con)o(tin)o(uous)h(on)g(])p Fo(\015)s(;)8 b(\030)r Fp([)k(and)h(so)g (lim)1463 2570 y Fk(z)q Fr(#)p Fk(\015)1529 2563 y Fo(g)r Fp(\()p Fo(z)r Fp(\))f(exists)129 2624 y(and)17 b(is)f(\014nite,)f(a)h (con)o(tradiction)g(with)g(the)g(minimali)o(t)o(y)d(of)j Fo(\015)s Fp(.)202 2684 y Fg(\(iv\))i Fn(9)p Fo(\021)e Fn(2)8 b Fp(]0)p Fo(;)g(\030)r Fp([)16 b(s.t.)21 b Fo(g)631 2666 y Fr(0)643 2684 y Fp(\()p Fo(z)r Fp(\))14 b Fn(6)p Fp(=)g(0,)i Fn(8)p Fo(z)e Fn(2)8 b Fp(]0)p Fo(;)g(\021)r Fp([.)926 2819 y(9)p eop %%Page: 10 10 10 9 bop 202 286 a Fp(Equalling)11 b(the)f(RHS)h(of)g(\(12)p (#equation.12) [[234 648 245 660] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (to)g(zero)f(w)o(e)f(get)h(a)h(quadratic)f(equation)g(with)g(resp)q (ect)129 347 y(to)16 b Fo(g)r Fp(\()p Fo(z)r Fp(\).)21 b(Its)c(solution)f(is)g(a)h(couple)f(of)g(functions,)445 476 y Fo(')477 483 y Fs(1)497 476 y Fp(\()p Fo(z)r Fp(\))d(=)748 443 y(2)p 630 465 261 2 v 630 514 a(1)f(+)715 475 y Fn(p)p 756 475 135 2 v 756 514 a Fp(1)g Fn(\000)e Fp(3)p Fo(z)896 476 y(;)56 b(')998 483 y Fs(2)1018 476 y Fp(\()p Fo(z)r Fp(\))14 b(=)1270 443 y(2)p 1151 465 262 2 v 1151 514 a(1)e Fn(\000)1237 475 y(p)p 1278 475 135 2 v 1278 514 a Fp(1)g Fn(\000)e Fp(3)p Fo(z)1418 476 y(;)129 615 y Fp(de\014ned)19 b(on)h(the)g(in)o(terv)m(al)e(]0)p Fo(;)8 b(\030)r Fp([)g Fn(\\)g Fp(]0)p Fo(;)850 596 y Fs(1)p 850 604 18 2 v 850 633 a(3)872 615 y Fp([.)31 b(Clearly)l(,)19 b Fo(')1146 597 y Fr(0)1146 628 y Fs(1)1166 615 y Fp(\()p Fo(z)r Fp(\))g Fo(>)h Fp(0)g(and)g Fo(')1480 597 y Fr(0)1480 628 y Fs(2)1500 615 y Fp(\()p Fo(z)r Fp(\))f Fo(<)h Fp(0)g(ev-)129 676 y(erywhere)d(on)i(that)f(in)o(terv)m(al.)26 b(This)19 b(implies)c(that)k(the)f(RHS)f(of)i(\(12)p (#equation.12) [[404 555 416 567] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g(v) m(anishes)f(in)g(a)129 736 y(p)q(oin)o(t)h Fo(z)i Fp(from)d(that)i(in)o (terv)m(al)e(if)h(and)g(only)g(if)g Fo(g)r Fp(\()p Fo(z)r Fp(\))g(=)g Fo(')1208 743 y Fs(1)1227 736 y Fp(\()p Fo(z)r Fp(\))g(or)h Fo(g)r Fp(\()p Fo(z)r Fp(\))f(=)f Fo(')1567 743 y Fs(2)1587 736 y Fp(\()p Fo(z)r Fp(\))h(and)129 796 y(in)g(suc)o(h)g(a)h(case)g(either)e(0)i(=)f Fo(g)720 778 y Fr(0)732 796 y Fp(\()p Fo(z)r Fp(\))h Fo(<)f(')904 778 y Fr(0)904 808 y Fs(1)924 796 y Fp(\()p Fo(z)r Fp(\))g(or)h(0)g(=)f Fo(g)1195 778 y Fr(0)1207 796 y Fp(\()p Fo(z)r Fp(\))g Fo(>)h(')1379 778 y Fr(0)1379 808 y Fs(2)1398 796 y Fp(\()p Fo(z)r Fp(\).)31 b(Th)o(us)20 b Fo(')1665 803 y Fs(1)1685 796 y Fp(\()p Fo(z)r Fp(\))129 856 y(coincides)d(with)h Fo(g)r Fp(\()p Fo(z)r Fp(\))h(in)f(at)h(most)f(one)g(p)q(oin)o(t)h Fo(z)r Fp(,)f(and)i(the)e(same)f(is)i(true)f(for)h Fo(')1652 863 y Fs(2)1671 856 y Fp(\()p Fo(z)r Fp(\).)129 916 y(Consequen)o(tly)l (,)14 b(there)i(exists)g Fo(\021)f Fn(2)8 b Fp(]0)p Fo(;)g(\030)r Fp([)g Fn(\\)g Fp(]0)p Fo(;)992 897 y Fs(1)p 992 905 V 992 934 a(3)1015 916 y Fp([)16 b(suc)o(h)g(that)g Fo(g)1285 898 y Fr(0)1297 916 y Fp(\()p Fo(z)r Fp(\))g(do)q(esn't)h(v)m(anish)f (on)129 977 y(]0)p Fo(;)8 b(\021)r Fp([.)20 b(Cho)q(ose)e Fo(\021)g Fp(ha)o(ving)e(this)g(prop)q(ert)o(y)l(.)202 1037 y Fg(\(v\))i Fo(g)307 1019 y Fr(0)319 1037 y Fp(\()p Fo(z)r Fp(\))c Fo(>)f Fp(0,)k Fn(8)p Fo(z)d Fn(2)8 b Fp(]0)p Fo(;)g(\021)r Fp([.)202 1097 y(If)14 b Fo(g)274 1079 y Fr(0)286 1097 y Fp(\()p Fo(z)r Fp(\))f Fo(<)h Fp(0,)h Fn(8)p Fo(z)f Fn(2)8 b Fp(]0)p Fo(;)g(\021)r Fp([,)14 b(then)h Fo(g)835 1104 y Fs(1)869 1097 y Fp(=)e(lim)988 1104 y Fk(z)q Fr(#)p Fs(0)1052 1097 y Fo(g)r Fp(\()p Fo(z)r Fp(\))h(exists)g(\(\014nite)g(or)h(in\014nite\))e(and)129 1157 y Fo(g)152 1164 y Fs(1)186 1157 y Fo(>)g(g)r Fp(\()p Fo(\021)r Fp(\))h Fo(>)g Fp(1.)22 b(On)16 b(the)g(other)g(hand,)h(in)f (virtue)f(of)h(\(12)p (#equation.12) [[349 439 360 451] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)573 1287 y Fo(z)598 1266 y Fs(2)618 1287 y Fo(g)643 1266 y Fr(0)655 1287 y Fp(\()p Fo(z)r Fp(\))d(=)h(1)e Fn(\000)905 1253 y Fp(1)p 873 1276 88 2 v 873 1321 a Fo(g)r Fp(\()p Fo(z)r Fp(\))977 1287 y Fn(\000)1032 1253 y Fp(3)p 1032 1276 25 2 v 1032 1321 a(4)1070 1287 y Fo(z)e(g)r Fp(\()p Fo(z)r Fp(\))k Fo(<)g Fp(0)8 b Fo(:)358 b Fp(\(15\))129 1428 y(Equalit)o(y)15 b(\(13)p (#equation.13) [[155 374 166 386] [1 1 1 [3 3]] [0 0 1]] pdfm (\))i (implies)c(that)k(\()p Fo(z)744 1410 y Fs(3)p Fk(=)p Fs(4)799 1428 y Fo(g)r Fp(\()p Fo(z)r Fp(\)\))906 1410 y Fr(0)931 1428 y Fo(>)d Fp(0)j(on)f(]0)p Fo(;)8 b(\021)r Fp([)16 b(and)h(so)584 1561 y Fo(g)r Fp(\()p Fo(z)r Fp(\))c Fo(<)h(g)r Fp(\()p Fo(\021)r Fp(\))834 1506 y Fl(\020)869 1528 y Fo(\021)p 869 1550 26 2 v 869 1596 a(z)900 1506 y Fl(\021)930 1517 y Fs(3)p Fk(=)p Fs(4)993 1561 y Fo(;)57 b Fn(8)p Fo(z)14 b Fn(2)8 b Fp(]0)p Fo(;)g(\021)r Fp([)g Fo(:)129 1685 y Fp(Consequen)o(tly)l(,)15 b(lim)508 1692 y Fk(z)q Fr(#)p Fs(0)572 1685 y Fo(z)10 b(g)r Fp(\()p Fo(z)r Fp(\))k(=)g(0.)21 b(Sending)16 b Fo(z)j Fp(to)d(0)h(in)f(\(15)p (#equation.15) [[365 313 376 325] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (giv)o(es)813 1816 y(1)11 b Fn(\000)912 1782 y Fp(1)p 903 1804 44 2 v 903 1850 a Fo(g)926 1857 y Fs(1)965 1816 y Fn(\024)i Fp(0)8 b Fo(;)129 1946 y Fp(a)16 b(con)o(tradiction.)202 2006 y Fg(\(vi\))i Fp(lim)365 2013 y Fk(z)q Fr(#)p Fs(0)428 2006 y Fo(g)r Fp(\()p Fo(z)r Fp(\))c(=)g(1.)202 2066 y(F)l(rom)k(Claim)f(\(v\))i(follo)o(ws)g(that)h Fo(g)857 2073 y Fs(1)896 2066 y Fp(=)e(lim)1020 2073 y Fk(z)q Fr(#)p Fs(0)1084 2066 y Fo(g)r Fp(\()p Fo(z)r Fp(\))h(exists)f(and)i(1) f Fn(\024)g Fo(g)1550 2073 y Fs(1)1588 2066 y Fo(<)g(g)r Fp(\()p Fo(\021)r Fp(\).)129 2127 y(Supp)q(ose)13 b(that)f Fo(g)440 2134 y Fs(1)474 2127 y Fo(>)i Fp(1.)20 b(Then)12 b(one)g(concludes)g(from)f(\(12)p (#equation.12) [[345 207 357 219] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (that)h(there)e(exists)h Fo(\016)j Fn(2)8 b Fp(]0)p Fo(;)g(\021)r Fp([)129 2187 y(s.t.)372 2311 y Fo(g)397 2290 y Fr(0)409 2311 y Fp(\()p Fo(z)r Fp(\))14 b Fo(>)552 2277 y(d)p 542 2299 45 2 v 542 2345 a(z)567 2330 y Fs(2)592 2311 y Fo(;)24 b Fn(8)p Fo(z)15 b Fn(2)8 b Fp(]0)p Fo(;)g(\016)r Fp([)g Fo(;)56 b Fp(where)16 b Fo(d)e Fp(=)1151 2277 y(1)p 1151 2299 25 2 v 1151 2345 a(2)1189 2240 y Fl(\022)1226 2311 y Fp(1)d Fn(\000)1325 2277 y Fp(1)p 1316 2299 44 2 v 1316 2345 a Fo(g)1339 2352 y Fs(1)1364 2240 y Fl(\023)1414 2311 y Fo(>)j Fp(0)p Fo(:)129 2444 y Fp(This)i(implies)590 2574 y Fo(g)r Fp(\()p Fo(z)r Fp(\))e Fo(<)f(g)r Fp(\()p Fo(\016)r Fp(\))e(+)895 2540 y Fo(d)p 895 2562 26 2 v 896 2608 a(\016)936 2574 y Fn(\000)991 2540 y Fo(d)p 991 2562 V 991 2608 a(z)1021 2574 y(;)25 b Fn(8)p Fo(z)14 b Fn(2)8 b Fp(]0)p Fo(;)g(\016)r Fp([)g Fo(;)129 2694 y Fp(a)16 b(con)o(tradiction.)p 1712 2694 2 33 v 1714 2663 30 2 v 1714 2694 V 1743 2694 2 33 v 914 2819 a(10)p eop %%Page: 11 11 11 10 bop 129 286 a Fh(Corollary)18 b(7.)24 b Fg(The)17 b(maximal)h(solution)g Fo(g)r Fp(\()p Fo(z)r Fp(\))g Fg(satis\014es)g(the)g(inte)n(gr)n(al)f(identity)398 408 y Fo(g)r Fp(\()p Fo(z)r Fp(\))486 387 y Fs(2)520 408 y Fp(=)d(2)p Fo(z)621 387 y Fr(\000)p Fs(3)p Fk(=)p Fs(2)712 340 y Fl(Z)761 353 y Fk(z)739 453 y Fs(0)790 408 y Fo(s)813 387 y Fr(\000)p Fs(1)p Fk(=)p Fs(2)895 408 y Fp(\()p Fo(g)r Fp(\()p Fo(s)p Fp(\))d Fn(\000)g Fp(1\))d Fo(ds;)59 b Fn(8)p Fo(z)14 b Fn(2)8 b Fp(]0)p Fo(;)g(z)1423 415 y Fs(0)1451 408 y Fp(])p Fo(:)182 b Fp(\(16\))129 530 y Fg(Pr)n(o)n(of.)22 b Fp(Rewrite)16 b(\(12)p (#equation.12) [[188 590 199 602] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h(as)671 625 y(\()p Fo(z)715 605 y Fs(3)p Fk(=)p Fs(2)770 625 y Fo(g)795 605 y Fs(2)815 625 y Fp(\))834 605 y Fr(0)860 625 y Fp(=)c(2)p Fo(z)960 605 y Fr(\000)p Fs(1)p Fk(=)p Fs(2)1043 625 y Fp(\()p Fo(g)g Fn(\000)e Fp(1\))p Fo(;)129 721 y Fp(and)17 b(in)o(tegrate)e(from)g(0)i(to)g Fo(z)h Fp(when)e(taking)g(in)o(to)g(accoun)o(t)h(\(14)p (#equation.14) [[375 544 386 556] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)p 1712 721 2 33 v 1714 690 30 2 v 1714 721 V 1743 721 2 33 v 129 863 a Ff(3.2)66 b(Asymptotics)30 b(of)f(the)g(left)h(maximal)h (solution)g Fe(g)r Fz(\()p Fe(z)s Fz(\))e Ff(at)282 938 y Fe(z)19 b Fz(=)d(0)129 1030 y Fh(Lemm)o(a)g(8.)24 b Fg(If)17 b Fo(\014)g Fn(\025)c Fp(0)18 b Fg(then)544 1156 y Fo(e)567 1135 y Fr(\000)599 1122 y Fj(1)p 599 1128 16 2 v 599 1148 a Fi(z)630 1088 y Fl(Z)680 1101 y Fs(1)658 1201 y Fk(z)708 1156 y Fo(s)731 1135 y Fk(\014)r Fr(\000)p Fs(2)800 1156 y Fo(e)828 1122 y Fj(1)p 828 1128 V 828 1148 a Fi(s)858 1156 y Fo(ds)c Fp(=)g Fo(O)q Fp(\()p Fo(z)1054 1135 y Fk(\014)1078 1156 y Fp(\))50 b Fg(as)16 b Fo(z)g Fn(#)e Fp(0)8 b Fo(:)129 1275 y Fg(Pr)n(o)n(of.)287 1412 y Fo(e)310 1391 y Fr(\000)342 1378 y Fj(1)p 342 1384 V 342 1404 a Fi(z)373 1344 y Fl(Z)423 1357 y Fs(1)401 1457 y Fk(z)451 1412 y Fo(s)474 1391 y Fk(\014)r Fr(\000)p Fs(2)543 1412 y Fo(e)571 1378 y Fj(1)p 570 1384 V 570 1404 a Fi(s)601 1412 y Fo(ds)42 b Fp(=)g Fo(e)794 1391 y Fr(\000)826 1378 y Fj(1)p 826 1384 V 826 1404 a Fi(z)857 1327 y Fl( )896 1344 y(Z)35 b Fj(1)p 951 1350 32 2 v 951 1370 a(2)p Fi(z)924 1457 y Fs(1)998 1412 y Fp(+)1044 1344 y Fl(Z)27 b Fj(1)p 1099 1350 16 2 v 1099 1370 a Fi(z)1084 1443 y Fj(1)p 1077 1449 32 2 v 1077 1470 a(2)p Fi(z)1122 1327 y Fl(!)1169 1412 y Fo(u)1197 1391 y Fr(\000)p Fk(\014)1248 1412 y Fo(e)1271 1391 y Fk(u)1301 1412 y Fo(du)691 1548 y Fn(\024)41 b Fo(e)794 1527 y Fr(\000)826 1514 y Fj(1)p 826 1520 16 2 v 826 1540 a Fi(z)857 1493 y Fl(\020)887 1548 y Fo(e)922 1514 y Fj(1)p 914 1520 32 2 v 914 1540 a(2)p Fi(z)964 1548 y Fn(\000)10 b Fo(e)1036 1493 y Fl(\021)1077 1548 y Fp(+)h(\(2)p Fo(z)r Fp(\))1213 1527 y Fk(\014)1237 1548 y Fo(e)1260 1527 y Fr(\000)1292 1514 y Fj(1)p 1292 1520 16 2 v 1292 1540 a Fi(z)1323 1493 y Fl(\020)1353 1548 y Fo(e)1381 1514 y Fj(1)p 1380 1520 V 1380 1540 a Fi(z)1414 1548 y Fn(\000)g Fo(e)1499 1514 y Fj(1)p 1491 1520 32 2 v 1491 1540 a(2)p Fi(z)1530 1493 y Fl(\021)1576 1548 y Fo(:)p 1712 1653 2 33 v 1714 1622 30 2 v 1714 1653 V 1743 1653 2 33 v 129 1746 a Fg(Notation.)25 b Fp(\()p Fo(\013)p Fp(\))423 1753 y Fs(0)456 1746 y Fp(=)14 b(1,)i(\()p Fo(\013)p Fp(\))631 1753 y Fk(j)664 1746 y Fp(=)e Fo(\013)p Fp(\()p Fo(\013)d Fp(+)g(1\))d Fo(:)g(:)g(:)h Fp(\()p Fo(\013)i Fp(+)g Fo(j)j Fn(\000)d Fp(1\).)129 1851 y Fh(Lemm)o(a)16 b(9.)24 b Fg(L)n(et)17 b Fo(\016)f(>)d Fp(0)p Fg(,)18 b Fo(\027)f Fn(2)d Fm(R)p Fg(.)20 b(Then)e(the)g(expr)n(ession)684 1978 y Fo(z)709 1957 y Fr(\000)p Fk(\027)758 1978 y Fo(e)781 1957 y Fr(\000)p Fs(1)p Fk(=z)871 1910 y Fl(Z)921 1923 y Fk(\016)899 2023 y(z)948 1978 y Fo(s)971 1957 y Fk(\027)r Fr(\000)p Fs(2)1038 1978 y Fo(e)1061 1957 y Fs(1)p Fk(=s)1122 1978 y Fo(ds)8 b(;)129 2097 y Fg(r)n(e)n(gar)n(de)n(d)15 b(as)i(a)h(function)h(in)e (the)h(variable)h Fo(z)r Fg(,)e(has)g(the)h(asymptotic)f(series,)h(as)f Fo(z)f Fn(#)e Fp(0)p Fg(,)846 2166 y Fr(1)828 2181 y Fl(X)833 2286 y Fk(j)r Fs(=0)900 2228 y Fp(\()p Fo(\027)s Fp(\))965 2235 y Fk(j)983 2228 y Fo(z)1008 2207 y Fk(j)1035 2228 y Fo(:)129 2371 y Fg(Pr)n(o)n(of.)22 b Fp(It)16 b(su\016ces)g(to)h(sho)o(w)g(that)f(it)g(holds,)g(for)h(all)e Fo(n)f Fn(2)g Fm(Z)1245 2378 y Fs(+)1271 2371 y Fp(,)348 2511 y Fo(e)371 2490 y Fr(\000)403 2476 y Fj(1)p 403 2482 16 2 v 403 2503 a Fi(z)434 2443 y Fl(Z)484 2456 y Fk(\016)462 2556 y(z)511 2511 y Fo(s)534 2490 y Fk(\027)r Fr(\000)p Fs(2)601 2511 y Fo(e)629 2476 y Fj(1)p 629 2482 V 629 2503 a Fi(s)659 2511 y Fo(ds)42 b Fp(=)f Fn(\000)p Fo(e)895 2476 y Fj(1)p 895 2482 V 895 2503 a Fi(\016)915 2490 y Fr(\000)947 2476 y Fj(1)p 947 2482 V 947 2503 a Fi(z)981 2448 y Fk(n)p Fr(\000)p Fs(1)979 2463 y Fl(X)984 2568 y Fk(j)r Fs(=0)1051 2511 y Fp(\()p Fo(\027)s Fp(\))1116 2518 y Fk(j)1134 2511 y Fo(\016)1158 2490 y Fk(\027)r Fs(+)p Fk(j)1234 2511 y Fp(+)1285 2448 y Fk(n)p Fr(\000)p Fs(1)1283 2463 y Fl(X)1288 2568 y Fk(j)r Fs(=0)1355 2511 y Fp(\()p Fo(\027)s Fp(\))1420 2518 y Fk(j)1438 2511 y Fo(z)1463 2490 y Fk(\027)r Fs(+)p Fk(j)828 2674 y Fp(+\()p Fo(\027)s Fp(\))931 2681 y Fk(n)955 2674 y Fo(e)978 2653 y Fr(\000)1010 2640 y Fj(1)p 1010 2646 V 1010 2666 a Fi(z)1041 2606 y Fl(Z)1091 2619 y Fk(\016)1069 2719 y(z)1118 2674 y Fo(s)1141 2653 y Fk(\027)r Fs(+)p Fk(n)p Fr(\000)p Fs(2)1257 2674 y Fo(e)1285 2640 y Fj(1)p 1284 2646 V 1284 2666 a Fi(s)1315 2674 y Fo(ds)8 b(:)914 2819 y Fp(11)p eop %%Page: 12 12 12 11 bop 129 286 a Fp(Actually)l(,)14 b(according)i(to)h(Lemma)d(8)p (#thm.8) [[261 648 267 660] [1 1 1 [3 3]] [0 0 1]] pdfm 17 w(this)i(means)f(that)i(the)f(relation)497 435 y Fo(z)522 414 y Fr(\000)p Fk(\027)571 435 y Fo(e)594 414 y Fr(\000)626 400 y Fj(1)p 626 406 16 2 v 626 427 a Fi(z)657 367 y Fl(Z)707 380 y Fk(\016)685 479 y(z)734 435 y Fo(s)757 414 y Fk(\027)r Fr(\000)p Fs(2)824 435 y Fo(e)852 400 y Fj(1)p 851 406 V 851 427 a Fi(s)882 435 y Fo(ds)e Fp(=)999 372 y Fk(n)p Fr(\000)p Fs(1)996 387 y Fl(X)1001 492 y Fk(j)r Fs(=0)1068 435 y Fp(\()p Fo(\027)s Fp(\))1133 442 y Fk(j)1151 435 y Fo(z)1176 414 y Fk(j)1205 435 y Fp(+)d Fo(O)q Fp(\()p Fo(z)1336 414 y Fk(n)1360 435 y Fp(\))129 588 y(holds)16 b(true)g(for)h(all)f Fo(n)d Fn(\025)h(\000)p Fo(\027)s Fp(,)i(and)h(so)g(for)f(all)g Fo(n)e Fn(2)g Fm(N)p Fp(.)202 648 y(W)l(e)j(shall)h(pro)q(ceed)g(b)o(y) f(induction)h(in)f Fo(n)p Fp(.)27 b(F)l(or)18 b Fo(n)f Fp(=)f(0)i(this)g(is)g(a)g(trivial)f(equalit)o(y)l(.)129 708 y(The)f(induction)g(step)g Fo(n)e Fn(!)g Fo(n)d Fp(+)g(1:)198 846 y Fo(e)221 825 y Fr(\000)253 811 y Fj(1)p 253 817 V 253 838 a Fi(z)284 778 y Fl(Z)334 791 y Fk(\016)312 890 y(z)361 846 y Fo(s)384 825 y Fk(\027)r Fs(+)p Fk(n)p Fr(\000)p Fs(2)500 846 y Fo(e)528 811 y Fj(1)p 527 817 V 527 838 a Fi(s)558 846 y Fo(ds)42 b Fp(=)f Fn(\000)p Fo(e)789 825 y Fr(\000)821 811 y Fj(1)p 821 817 V 821 838 a Fi(z)852 778 y Fl(Z)902 791 y Fk(\016)880 890 y(z)929 846 y Fo(s)952 825 y Fk(\027)r Fs(+)p Fk(n)1031 790 y Fl(\020)1060 846 y Fo(e)1088 811 y Fj(1)p 1088 817 V 1088 838 a Fi(s)1110 790 y Fl(\021)1140 801 y Fr(0)1168 846 y Fo(ds)648 985 y Fp(=)g Fn(\000)p Fo(e)789 964 y Fr(\000)821 951 y Fj(1)p 821 957 V 821 977 a Fi(z)844 985 y Fp([)p Fo(s)881 964 y Fk(\027)r Fs(+)p Fk(n)951 985 y Fo(e)979 951 y Fj(1)p 978 957 V 978 977 a Fi(s)1001 985 y Fp(])1015 964 y Fk(\016)1015 997 y(z)1045 985 y Fp(+)11 b(\()p Fo(\027)j Fp(+)d Fo(n)p Fp(\))p Fo(e)1271 964 y Fr(\000)1303 951 y Fj(1)p 1303 957 V 1303 977 a Fi(z)1335 917 y Fl(Z)1384 930 y Fk(\016)1362 1030 y(z)1412 985 y Fo(s)1435 964 y Fk(\027)r Fs(+)p Fk(n)p Fr(\000)p Fs(1)1550 985 y Fo(e)1578 951 y Fj(1)p 1578 957 V 1578 977 a Fi(s)1608 985 y Fo(ds)d(:)p 1712 1112 2 33 v 1714 1081 30 2 v 1714 1112 V 1743 1112 2 33 v 129 1209 a Fp(Let)17 b(us)g(consider)g(equation)f(\(11)p (#equation.11) [[237 427 249 439] [1 1 1 [3 3]] [0 0 1]] pdfm (\))i (\(without)f(the)g(initial)e(condition\))i(in)g(the)f(space)h(of)129 1269 y(formal)e(p)q(o)o(w)o(er)h(series)g Fm(C)9 b Fp([[)p Fo(z)r Fp(]].)22 b(Its)16 b(solution)686 1413 y(~)-26 b Fo(g)r Fp(\()p Fo(z)r Fp(\))14 b(=)856 1350 y Fr(1)838 1365 y Fl(X)841 1472 y Fk(k)q Fs(=0)918 1413 y Fo(\013)949 1420 y Fk(k)970 1413 y Fo(z)995 1392 y Fk(k)1030 1413 y Fn(2)g Fm(C)9 b Fp([[)p Fo(z)r Fp(]])470 b(\(17\))129 1561 y(is)16 b(unique,)f(with)h(the)g(co)q(e\016cien)o(ts)f(b)q(eing)h (determined)e(b)o(y)i(the)g(recursiv)o(e)e(relation)388 1711 y Fo(\013)419 1718 y Fs(0)453 1711 y Fp(=)f(1)p Fo(;)57 b(\013)630 1718 y Fk(k)q Fs(+1)711 1711 y Fp(=)762 1640 y Fl(\022)804 1677 y Fp(1)p 804 1699 25 2 v 804 1745 a(2)833 1711 y Fo(k)14 b Fp(+)925 1677 y(3)p 925 1699 V 925 1745 a(4)955 1640 y Fl(\023)1026 1648 y Fk(k)1000 1663 y Fl(X)1005 1768 y Fk(j)r Fs(=0)1080 1711 y Fo(\013)1111 1718 y Fk(j)1129 1711 y Fo(\013)1160 1718 y Fk(k)q Fr(\000)p Fk(j)1274 1711 y Fp(for)j Fo(k)f Fn(\025)d Fp(0)8 b Fo(:)173 b Fp(\(18\))129 1864 y(Sev)o(eral)15 b(\014rst)h(co)q(e\016cien)o(ts)f (are)493 1980 y Fo(\013)524 1987 y Fs(0)558 1980 y Fp(=)f(1)p Fo(;)24 b(\013)703 1987 y Fs(1)737 1980 y Fp(=)794 1946 y(3)p 794 1968 V 794 2014 a(4)823 1980 y Fo(;)g(\013)892 1987 y Fs(2)926 1980 y Fp(=)982 1946 y(15)p 982 1968 49 2 v 995 2014 a(8)1036 1980 y Fo(;)g(\013)1105 1987 y Fs(3)1139 1980 y Fp(=)1196 1946 y(483)p 1196 1968 74 2 v 1208 2014 a(64)1274 1980 y Fo(;)8 b(:)g(:)g(:)15 b(:)129 2139 y Fh(Prop)r(osition)i(10.)24 b Fg(The)c(left)g(maximal)f (solution)h Fo(g)r Fp(\()p Fo(z)r Fp(\))f Fg(has)g(an)h(asymptotic)e (series,)129 2199 y(as)f Fo(z)f Fn(#)d Fp(0)p Fg(,)18 b(that)g(is)f(e)n(qual)h(to)856 2278 y Fr(1)838 2293 y Fl(X)842 2399 y Fk(k)q Fs(=0)918 2340 y Fo(\013)949 2347 y Fk(k)970 2340 y Fo(z)995 2319 y Fk(k)1025 2340 y Fo(:)129 2488 y Fg(Pr)n(o)n(of.)k Fp(W)l(e)16 b(ha)o(v)o(e)g(to)g (sho)o(w)h(that,)g(for)f(all)g Fo(n)e Fn(2)g Fm(Z)1046 2495 y Fs(+)1073 2488 y Fp(,)676 2630 y Fo(g)r Fp(\()p Fo(z)r Fp(\))g(=)855 2568 y Fk(n)830 2583 y Fl(X)833 2689 y Fk(k)q Fs(=0)910 2630 y Fo(\013)941 2637 y Fk(k)962 2630 y Fo(z)987 2610 y Fk(k)1020 2630 y Fp(+)d Fo(o)p Fp(\()p Fo(z)1136 2610 y Fk(n)1159 2630 y Fp(\))d Fo(:)461 b Fp(\(19\))914 2819 y(12)p eop %%Page: 13 13 13 12 bop 129 286 a Fp(W)l(e)13 b(shall)g(pro)q(ceed)h(b)o(y)f (induction)g(in)h Fo(n)p Fp(.)20 b(The)14 b(case)f Fo(n)h Fp(=)g(0)g(means)f(that)h(lim)1544 293 y Fk(z)q Fr(#)p Fs(0)1608 286 y Fo(g)r Fp(\()p Fo(z)r Fp(\))g(=)129 347 y(1)h(and)g(is)g(co)o(v)o(ered)f(b)o(y)g(Prop)q(osition)i(6)p (#thm.6) [[266 634 272 646] [1 1 1 [3 3]] [0 0 1]] pdfm (.)21 b(Let)15 b(us)g(supp)q(ose)h(that)g(\(19)p (#equation.19) [[389 634 401 646] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f (is)g(v)m(alid)g(for)g(some)129 407 y Fo(n)f Fn(2)g Fm(Z)255 414 y Fs(+)281 407 y Fp(.)22 b(Denote)16 b(\(in)g(this)g(pro)q(of)s(\)) 603 548 y Fo(a)629 555 y Fr(\000)672 548 y Fp(=)e(lim)8 b(inf)765 580 y Fk(z)q Fr(#)p Fs(0)872 514 y Fo(g)r Fp(\()p Fo(z)r Fp(\))j Fn(\000)1021 477 y Fl(P)1073 490 y Fk(n)1073 529 y(k)q Fs(=0)1148 514 y Fo(\013)1179 521 y Fk(k)1200 514 y Fo(z)1225 496 y Fk(k)p 872 537 375 2 v 1013 582 a Fo(z)1038 568 y Fk(n)p Fs(+1)1260 548 y Fo(:)129 680 y Fp(Similarly)l(,)16 b Fo(a)375 687 y Fs(+)423 680 y Fp(designates)k(the)f(limes)e(sup)q(erior,)j(as)f Fo(z)i Fn(#)e Fp(0,)h(of)f(the)g(same)f(function.)129 740 y(Th)o(us)h(the)f (induction)g(step)h Fo(n)f Fn(!)g Fo(n)13 b Fp(+)f(1)20 b(means)d(to)i(v)o(erify)e(that)i Fo(a)1402 747 y Fr(\000)1450 740 y Fp(=)e Fo(a)1531 747 y Fs(+)1579 740 y Fp(=)g Fo(\013)1665 747 y Fk(n)p Fs(+1)1734 740 y Fp(.)129 800 y(W)l(e)f(shall)g(do)g(it)g (in)g(three)g(steps.)202 860 y Fg(\(i\))i Fo(a)301 867 y Fr(\000)344 860 y Fn(\024)13 b Fo(\013)427 867 y Fk(n)p Fs(+1)510 860 y Fn(\024)g Fo(a)588 867 y Fs(+)617 860 y Fp(.)202 920 y(If)d Fo(b)k(<)g(a)358 927 y Fr(\000)398 920 y Fp(then)d(there)f(exists)h Fo(\016)k(>)f Fp(0)e(s.t.)19 b Fo(g)r Fp(\()p Fo(z)r Fp(\))14 b Fo(>)1116 883 y Fl(P)1169 896 y Fk(n)1169 935 y(k)q Fs(=0)1244 920 y Fo(\013)1275 927 y Fk(k)1296 920 y Fo(z)1321 902 y Fk(k)1343 920 y Fp(+)q Fo(b)8 b(z)1436 902 y Fk(n)p Fs(+1)1504 920 y Fp(,)j Fn(8)p Fo(z)k Fn(2)8 b Fp(]0)p Fo(;)g(\016)r Fp([.)129 981 y(F)l(urther,)15 b(from)g(the)h(recurren)o(t)f(relation)h(\(18)p (#equation.18) [[294 482 306 494] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (one)f(\014nds)h(that)407 1137 y(~)-25 b Fo(g)r Fp(\()p Fo(z)r Fp(\))494 1117 y Fs(2)527 1137 y Fp(=)597 1075 y Fr(1)579 1090 y Fl(X)583 1196 y Fk(k)q Fs(=0)659 1052 y Fl( )725 1075 y Fk(k)699 1090 y Fl(X)704 1195 y Fk(j)r Fs(=0)779 1137 y Fo(\013)810 1144 y Fk(j)828 1137 y Fo(\013)859 1144 y Fk(k)q Fr(\000)p Fk(j)924 1052 y Fl(!)972 1137 y Fo(z)997 1117 y Fk(k)1032 1137 y Fp(=)1102 1075 y Fr(1)1084 1090 y Fl(X)1087 1196 y Fk(k)q Fs(=0)1214 1104 y Fp(2)p 1169 1126 115 2 v 1169 1173 a Fo(k)13 b Fp(+)1261 1153 y Fs(3)p 1261 1161 18 2 v 1261 1190 a(2)1297 1137 y Fo(\013)1328 1144 y Fk(k)q Fs(+1)1394 1137 y Fo(z)1419 1117 y Fk(k)1449 1137 y Fo(;)198 b Fp(\(20\))129 1300 y(and)17 b(the)f(assumption)h (\(19)p (#equation.19) [[213 405 224 417] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g(implies)d(that)j(the)g(asymptotics)e(of)i Fo(g)r Fp(\()p Fo(z)r Fp(\))1441 1282 y Fs(2)1478 1300 y Fp(is)f(giv)o(en)g(b) o(y)g(a)129 1360 y(truncation)g(of)h(the)f(formal)f(p)q(o)o(w)o(er)h (series)h(~)-25 b Fo(g)r Fp(\()p Fo(z)r Fp(\))1021 1342 y Fs(2)1040 1360 y Fp(,)16 b(namely)581 1508 y Fo(g)r Fp(\()p Fo(z)r Fp(\))669 1487 y Fs(2)703 1508 y Fp(=)780 1446 y Fk(n)755 1461 y Fl(X)758 1567 y Fk(k)q Fs(=0)885 1474 y Fp(2)p 840 1497 115 2 v 840 1544 a Fo(k)d Fp(+)932 1524 y Fs(3)p 932 1532 18 2 v 932 1561 a(2)960 1508 y Fo(\013)991 1515 y Fk(k)q Fs(+1)1057 1508 y Fo(z)1082 1487 y Fk(k)1114 1508 y Fp(+)e Fo(o)p Fp(\()p Fo(z)1230 1487 y Fk(n)1254 1508 y Fp(\))d Fo(:)129 1663 y Fp(Com)o(bining)15 b(this)h(with)g(\(16)p (#equation.16) [[216 318 228 330] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (leads)f(to)h(the)f(conclusion)g(that,)g Fn(8)p Fo(z)e Fn(2)8 b Fp(]0)p Fo(;)g(\016)r Fp([,)219 1754 y Fk(n)194 1769 y Fl(X)197 1875 y Fk(k)q Fs(=0)324 1783 y Fp(2)p 279 1805 115 2 v 279 1852 a Fo(k)13 b Fp(+)371 1833 y Fs(3)p 371 1841 18 2 v 371 1869 a(2)407 1817 y Fo(\013)438 1824 y Fk(k)q Fs(+1)504 1817 y Fo(z)529 1796 y Fk(k)562 1817 y Fp(+)e Fo(o)p Fp(\()p Fo(z)678 1796 y Fk(n)701 1817 y Fp(\))42 b Fo(>)f Fp(2)8 b Fo(z)898 1796 y Fr(\000)p Fs(3)p Fk(=)p Fs(2)990 1749 y Fl(Z)1039 1762 y Fk(z)1017 1862 y Fs(0)1068 1817 y Fo(s)1091 1796 y Fr(\000)p Fs(1)p Fk(=)p Fs(2)1181 1732 y Fl( )1246 1754 y Fk(n)1221 1769 y Fl(X)1225 1875 y Fk(k)q Fs(=1)1301 1817 y Fo(\013)1332 1824 y Fk(k)1353 1817 y Fo(s)1376 1796 y Fk(k)1409 1817 y Fp(+)j Fo(b)d(s)1510 1796 y Fk(n)p Fs(+1)1578 1732 y Fl(!)1634 1817 y Fo(ds)762 1979 y Fp(=)41 b(2)899 1916 y Fk(n)874 1931 y Fl(X)878 2037 y Fk(k)q Fs(=1)990 1945 y Fo(\013)1021 1952 y Fk(k)p 959 1967 115 2 v 959 2014 a Fo(k)13 b Fp(+)1051 1995 y Fs(1)p 1051 2003 18 2 v 1051 2031 a(2)1087 1979 y Fo(z)1112 1958 y Fk(k)q Fr(\000)p Fs(1)1190 1979 y Fp(+)e(2)1324 1945 y Fo(b)p 1276 1967 117 2 v 1276 2014 a(n)g Fp(+)1370 1995 y Fs(3)p 1370 2003 18 2 v 1370 2031 a(2)1406 1979 y Fo(z)1431 1958 y Fk(n)1463 1979 y Fo(:)129 2131 y Fp(Hence)16 b Fo(b)f Fn(\024)g Fo(\013)396 2138 y Fk(n)p Fs(+1)482 2131 y Fp(for)i(all)g Fo(b)e(<)h(a)742 2138 y Fr(\000)788 2131 y Fp(and)i(consequen)o(tly)e Fo(a)1200 2138 y Fr(\000)1244 2131 y Fn(\024)g Fo(\013)1330 2138 y Fk(n)p Fs(+1)1398 2131 y Fp(.)24 b(The)18 b(inequalit)o(y)129 2191 y Fo(a)155 2198 y Fs(+)198 2191 y Fn(\025)13 b Fo(\013)281 2198 y Fk(n)p Fs(+1)366 2191 y Fp(can)j(b)q(e)h(pro)o(v)o(en)e(symmetrical)o (ly)l(.)202 2251 y Fg(\(ii\))j Fp(Let)g Fo(\025)d(>)g Fp(0)i(b)q(e)h(a)f(\014xed)f(parameter.)23 b(In)16 b(the)h(second)g (step)g(w)o(e)g(in)o(tro)q(duce)f(an)129 2311 y(auxiliary)e(function)h Fo(')p Fp(\()p Fo(z)r Fp(\))f(whose)i(c)o(hoice)e(dep)q(ends)h(on)h (whether)e Fo(n)g Fp(=)g(0)i(or)f Fo(n)f(>)g Fp(0.)21 b(In)129 2372 y(the)16 b(former)e(case)j(w)o(e)f(set)573 2502 y Fo(')p Fp(\()p Fo(z)r Fp(\))41 b(=)795 2468 y(1)p 793 2490 29 2 v 793 2536 a Fo(\025)827 2502 y(z)852 2481 y Fr(\000)884 2468 y Fj(3)p 884 2474 16 2 v 884 2494 a(4)906 2502 y Fo(e)929 2481 y Fr(\000)971 2468 y Fj(1)p 961 2474 35 2 v 961 2494 a Fi(\025z)1011 2434 y Fl(Z)1061 2447 y Fs(1)1039 2547 y Fk(z)1089 2502 y Fo(s)1112 2481 y Fr(\000)1144 2468 y Fj(5)p 1144 2474 16 2 v 1144 2494 a(4)1166 2502 y Fo(e)1203 2468 y Fj(1)p 1194 2474 33 2 v 1194 2494 a Fi(\025s)1242 2502 y Fo(ds)709 2641 y Fp(=)g(\()p Fo(\025z)r Fp(\))879 2620 y Fr(\000)911 2607 y Fj(3)p 912 2613 16 2 v 912 2633 a(4)934 2641 y Fo(e)957 2620 y Fr(\000)999 2607 y Fj(1)p 989 2613 35 2 v 989 2633 a Fi(\025z)1039 2573 y Fl(Z)1089 2586 y Fk(\025)1066 2686 y(\025z)1120 2641 y Fo(s)1143 2620 y Fr(\000)1175 2607 y Fj(5)p 1175 2613 16 2 v 1175 2633 a(4)1197 2641 y Fo(e)1225 2607 y Fj(1)p 1225 2613 V 1225 2633 a Fi(s)1255 2641 y Fo(ds)358 b Fp(\(21\))914 2819 y(13)p eop %%Page: 14 14 14 13 bop 129 286 a Fp(and)17 b(in)e(the)h(latter)g(one)537 429 y Fo(')p Fp(\()p Fo(z)r Fp(\))d(=)h Fo(z)722 408 y Fr(\000)754 395 y Fj(3)p 754 401 16 2 v 754 421 a(4)777 429 y Fo(e)800 408 y Fr(\000)832 395 y Fj(1)p 832 401 V 832 421 a Fi(z)863 361 y Fl(Z)913 374 y Fs(1)891 474 y Fk(z)944 367 y(n)p Fs(+1)941 382 y Fl(X)945 488 y Fk(k)q Fs(=0)1021 429 y Fo(')1053 436 y Fk(k)1083 429 y Fo(s)1111 395 y Fj(3)p 1111 401 V 1111 421 a(4)1131 408 y Fs(+)p Fk(k)q Fr(\000)p Fs(2)1233 429 y Fo(e)1261 395 y Fj(1)p 1261 401 V 1261 421 a Fi(s)1291 429 y Fo(ds)322 b Fp(\(22\))129 581 y(where)522 677 y Fo(')554 684 y Fs(0)587 677 y Fp(=)14 b Fo(\013)670 684 y Fs(0)690 677 y Fo(;)520 784 y(')552 791 y Fk(k)587 784 y Fp(=)g Fo(\013)670 791 y Fk(k)702 784 y Fn(\000)752 714 y Fl(\022)789 784 y Fo(k)f Fn(\000)882 751 y Fp(1)p 882 773 25 2 v 882 818 a(4)911 714 y Fl(\023)956 784 y Fo(\013)987 791 y Fk(k)q Fr(\000)p Fs(1)1102 784 y Fp(for)k(1)d Fn(\024)f Fo(k)j Fn(\024)e Fo(n;)473 920 y(')505 927 y Fk(n)p Fs(+1)587 920 y Fp(=)g Fo(\025)8 b(\013)706 927 y Fk(n)p Fs(+1)786 920 y Fn(\000)836 850 y Fl(\022)873 920 y Fo(n)j Fp(+)967 887 y(3)p 967 909 V 967 955 a(4)996 850 y Fl(\023)1041 920 y Fo(\013)1072 927 y Fk(n)1096 920 y Fo(:)1661 814 y Fp(\(23\))202 1063 y(Observ)o(e)20 b(that)h Fo(')536 1070 y Fs(1)577 1063 y Fp(=)h Fo(\013)668 1070 y Fs(1)702 1063 y Fn(\000)760 1043 y Fs(3)p 760 1051 18 2 v 760 1080 a(4)783 1063 y Fo(\013)814 1070 y Fs(0)855 1063 y Fp(=)g(0.)35 b(In)21 b(the)f(case)h Fo(n)h Fp(=)g(0,)g Fo(')p Fp(\()p Fo(z)r Fp(\))e(solv)o(es)h(the)129 1123 y(di\013eren)o(tial)15 b(equation)582 1258 y Fo(')614 1237 y Fr(0)626 1258 y Fp(\()p Fo(z)r Fp(\))f(=)788 1224 y(1)p 759 1246 82 2 v 759 1292 a Fo(\025)8 b(z)820 1277 y Fs(2)846 1258 y Fp(\()p Fo(')p Fp(\()p Fo(z)r Fp(\))j Fn(\000)f Fp(1\))i Fn(\000)1130 1224 y Fp(3)p 1130 1246 25 2 v 1130 1292 a(4)1164 1224 y Fo(')p Fp(\()p Fo(z)r Fp(\))p 1164 1246 95 2 v 1199 1292 a Fo(z)1272 1258 y(;)375 b Fp(\(24\))129 1383 y(while)15 b(in)h(the)g(case)g Fo(n)e(>)g Fp(0,)i Fo(')p Fp(\()p Fo(z)r Fp(\))g(solv)o(es)462 1538 y Fo(')494 1518 y Fr(0)506 1538 y Fp(\()p Fo(z)r Fp(\))d(=)649 1505 y(1)p 639 1527 45 2 v 639 1572 a Fo(z)664 1558 y Fs(2)689 1538 y Fp(\()p Fo(')p Fp(\()p Fo(z)r Fp(\))d Fn(\000)h Fp(1\))h Fn(\000)973 1505 y Fp(3)p 973 1527 25 2 v 973 1572 a(4)1007 1505 y Fo(')p Fp(\()p Fo(z)r Fp(\))p 1007 1527 95 2 v 1042 1572 a Fo(z)1118 1538 y Fn(\000)1170 1476 y Fk(n)p Fs(+1)1168 1491 y Fl(X)1171 1597 y Fk(k)q Fs(=2)1248 1538 y Fo(')1280 1545 y Fk(k)1301 1538 y Fo(z)1326 1518 y Fk(k)q Fr(\000)p Fs(2)1401 1538 y Fo(:)246 b Fp(\(25\))129 1693 y(W)l(e)14 b(claim)e(that,)j(in)f(the)g(b)q(oth)h(cases,)g(the)f (asymptotic)f(b)q(eha)o(viour)h(of)h Fo(')p Fp(\()p Fo(z)r Fp(\),)f(as)h Fo(z)h Fn(#)e Fp(0,)129 1754 y(is)i(giv)o(en)f(b)o(y)505 1899 y Fo(')p Fp(\()p Fo(z)r Fp(\))f(=)691 1836 y Fk(n)666 1851 y Fl(X)669 1957 y Fk(k)q Fs(=0)746 1899 y Fo(\013)777 1906 y Fk(k)798 1899 y Fo(z)823 1878 y Fk(k)856 1899 y Fp(+)d Fo(\025)d(\013)972 1906 y Fk(n)p Fs(+1)1041 1899 y Fo(z)1066 1878 y Fk(n)p Fs(+1)1145 1899 y Fp(+)j Fo(o)p Fp(\()p Fo(z)1261 1878 y Fk(n)p Fs(+1)1330 1899 y Fp(\))d Fo(:)290 b Fp(\(26\))129 2054 y(Actually)l(,)10 b(equalit)o(y)f(\(26)p (#equation.26) [[199 224 211 236] [1 1 1 [3 3]] [0 0 1]] pdfm (\))j (follo)o(ws)e(directly)f(from)h(Lemma)f(9)p (#thm.9) [[364 224 369 236] [1 1 1 [3 3]] [0 0 1]] pdfm (.)20 b(In)10 b(more)g(detail,)h(Lemma)129 2114 y(9)p (#thm.9) [[103 210 109 222] [1 1 1 [3 3]] [0 0 1]] pdfm 16 w(giv)o(es,)k(for)i(the)f(case)g Fo(n)e(>)g Fp(0,)428 2268 y Fo(')p Fp(\()p Fo(z)r Fp(\))41 b(=)647 2206 y Fk(n)p Fs(+1)644 2221 y Fl(X)648 2327 y Fk(k)q Fs(=0)724 2268 y Fo(')756 2275 y Fk(k)777 2268 y Fo(z)802 2247 y Fk(k)832 2206 y(n)p Fs(+1)p Fr(\000)p Fk(k)852 2221 y Fl(X)858 2326 y Fk(j)r Fs(=0)953 2198 y Fl(\022)990 2268 y Fo(k)13 b Fp(+)1082 2234 y(3)p 1082 2256 25 2 v 1082 2302 a(4)1111 2198 y Fl(\023)1148 2317 y Fk(j)1175 2268 y Fo(z)1200 2247 y Fk(j)1229 2268 y Fp(+)e Fo(o)p Fp(\()p Fo(z)1345 2247 y Fk(n)p Fs(+1)1414 2268 y Fp(\))564 2442 y(=)649 2380 y Fk(n)p Fs(+1)646 2395 y Fl(X)644 2500 y Fk(m)p Fs(=0)749 2380 y Fk(m)728 2395 y Fl(X)732 2501 y Fk(k)q Fs(=0)809 2442 y Fo(')841 2449 y Fk(k)870 2372 y Fl(\022)907 2442 y Fo(k)i Fp(+)999 2409 y(3)p 999 2431 V 999 2476 a(4)1028 2372 y Fl(\023)1065 2492 y Fk(m)p Fr(\000)p Fk(k)1153 2442 y Fo(z)1178 2422 y Fk(m)1223 2442 y Fp(+)e Fo(o)p Fp(\()p Fo(z)1339 2422 y Fk(n)p Fs(+1)1407 2442 y Fp(\))d Fo(:)213 b Fp(\(27\))129 2597 y(The)11 b(co)q(e\016cien)o(ts)f Fo(')498 2604 y Fk(k)520 2597 y Fp(,)i(as)g(giv)o(en)f(in)g(\(23)p (#equation.23) [[263 94 274 106] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)i (ha)o(v)o(e)d(b)q(een)i(c)o(hosen)f(so)h(that)g(the)g(asymptotics)129 2658 y(\(26)p (#equation.26) [[108 79 119 91] [1 1 1 [3 3]] [0 0 1]] pdfm (\))i(is)f (satis\014ed.)20 b(This)14 b(is)f(to)g(sa)o(y)h(that)f(equalling)f (\(27)p (#equation.27) [[336 79 347 91] [1 1 1 [3 3]] [0 0 1]] pdfm (\))i(to)g(\(26)p (#equation.26) [[373 79 385 91] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (leads)f(to)h(a)g(system)d(of)914 2819 y(14)p eop %%Page: 15 15 15 14 bop 129 286 a Fp(linear)16 b(equations)h(on)g(the)g(co)q (e\016cien)o(ts)e Fo(')916 293 y Fk(k)954 286 y Fp(whose)j(unique)e (solution)h(is)f(exactly)g(\(23)p (#equation.23) [[475 648 487 660] [1 1 1 [3 3]] [0 0 1]] pdfm (\))129 347 y(as)h(follo)o(ws)f(from)f(the)h(iden)o(tit)o(y)304 424 y Fk(m)283 439 y Fl(X)287 545 y Fk(k)q Fs(=0)364 486 y Fo(\013)395 493 y Fk(k)433 416 y Fl(\022)469 486 y Fo(k)d Fp(+)561 452 y(3)p 561 475 25 2 v 561 520 a(4)591 416 y Fl(\023)627 535 y Fk(m)p Fr(\000)p Fk(k)718 486 y Fn(\000)789 424 y Fk(m)768 439 y Fl(X)772 545 y Fk(k)q Fs(=1)848 486 y Fo(\013)879 493 y Fk(k)q Fr(\000)p Fs(1)963 416 y Fl(\022)999 486 y Fo(k)g Fn(\000)1092 452 y Fp(1)p 1092 475 V 1092 520 a(4)1121 416 y Fl(\023)k(\022)1211 486 y Fo(k)c Fp(+)1303 452 y(3)p 1303 475 V 1303 520 a(4)1333 416 y Fl(\023)1370 535 y Fk(m)p Fr(\000)p Fk(k)1463 486 y Fp(=)h Fo(\013)1546 493 y Fk(m)1579 486 y Fo(:)129 632 y Fp(The)i(case)g Fo(n)e Fp(=)g(0)j(is)f(ev)o(en)f(more)g(straigh)o (tforw)o(ard.)202 693 y Fg(\(iii\))j Fo(a)331 700 y Fs(+)374 693 y Fn(\024)13 b Fo(\013)457 700 y Fk(n)p Fs(+1)542 693 y Fp(and)k Fo(a)663 700 y Fr(\000)706 693 y Fn(\025)c Fo(\013)789 700 y Fk(n)p Fs(+1)858 693 y Fp(.)202 753 y(Let)i(us)g(sho)o(w)g(the)g(\014rst)g(inequalit)o(y)l(,)d(the)i(other) h(one)g(can)g(b)q(e)g(pro)o(v)o(en)f(analogously)l(.)129 813 y(W)l(e)i(shall)g(need)g(the)g(asymptotics)f(of)h(the)g(function) 248 933 y(\()p Fo(g)r Fp(\()p Fo(z)r Fp(\))11 b Fn(\000)g Fp(1\))459 914 y Fs(2)p 248 955 231 2 v 293 1000 a Fo(g)r Fp(\()p Fo(z)r Fp(\))d Fo(z)414 986 y Fs(2)525 966 y Fp(=)605 881 y Fl( )669 904 y Fk(n)644 919 y Fl(X)648 1025 y Fk(k)q Fs(=0)724 966 y Fo(\013)755 973 y Fk(k)777 966 y Fo(z)802 946 y Fk(k)834 966 y Fp(+)j Fo(o)p Fp(\()p Fo(z)950 946 y Fk(n)974 966 y Fp(\))993 881 y Fl(!)1032 892 y Fr(\000)p Fs(1)1088 881 y Fl( )1152 904 y Fk(n)1127 919 y Fl(X)1131 1025 y Fk(k)q Fs(=1)1207 966 y Fo(\013)1238 973 y Fk(k)1260 966 y Fo(z)1285 946 y Fk(k)q Fr(\000)p Fs(1)1362 966 y Fp(+)g Fo(o)p Fp(\()p Fo(z)1478 946 y Fk(n)p Fr(\000)p Fs(1)1547 966 y Fp(\))1566 881 y Fl(!)1605 892 y Fs(2)525 1135 y Fp(=)607 1073 y Fk(n)p Fr(\000)p Fs(1)605 1088 y Fl(X)609 1194 y Fk(k)q Fs(=0)685 1135 y Fo(\015)710 1142 y Fk(k)732 1135 y Fo(z)757 1115 y Fk(k)789 1135 y Fp(+)g Fo(o)p Fp(\()p Fo(z)905 1115 y Fk(n)p Fr(\000)p Fs(1)974 1135 y Fp(\))d Fo(:)129 1293 y Fp(Again,)15 b(assumption)h(\(19)p (#equation.19) [[207 407 218 419] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (implies)d(that)916 1255 y Fl(P)969 1269 y Fk(n)p Fr(\000)p Fs(1)969 1307 y Fk(k)q Fs(=0)1046 1293 y Fo(\015)1071 1300 y Fk(k)1092 1293 y Fo(z)1117 1275 y Fk(k)1155 1293 y Fp(is)h(a)i(truncation)f(of)h(the)f(p)q(o)o(w)o(er)129 1353 y(series)516 1441 y(\()p Fl(e)-28 b Fo(g)r Fp(\()p Fo(z)r Fp(\))11 b Fn(\000)g Fp(1\))727 1422 y Fs(2)p 516 1463 V 561 1508 a Fl(e)-28 b Fo(g)r Fp(\()p Fo(z)r Fp(\))8 b Fo(z)682 1494 y Fs(2)794 1474 y Fp(=)41 b(\()r(~)-26 b Fo(g)13 b Fn(\000)e Fp(1\))r(~)-26 b Fo(g)1046 1454 y Fr(0)1069 1474 y Fp(+)1123 1441 y(3)p 1123 1463 25 2 v 1123 1508 a(4)1159 1441 y(~)h Fo(g)r Fp(\()r(~)f Fo(g)13 b Fn(\000)e Fp(1\))p 1158 1463 174 2 v 1232 1508 a Fo(z)794 1611 y Fp(=)878 1577 y(1)p 878 1599 25 2 v 878 1645 a(2)907 1611 y(\()r(~)-26 b Fo(g)951 1590 y Fs(2)971 1611 y Fp(\))990 1590 y Fr(0)1013 1611 y Fn(\000)13 b Fp(~)-26 b Fo(g)1088 1590 y Fr(0)1111 1611 y Fp(+)1165 1577 y(3)p 1165 1599 V 1165 1645 a(4)1209 1577 y(~)g Fo(g)1232 1559 y Fs(2)1263 1577 y Fn(\000)13 b Fp(~)-26 b Fo(g)p 1207 1599 131 2 v 1260 1645 a(z)1351 1611 y(:)129 1730 y Fp(Here)16 b(w)o(e)h(ha)o(v)o(e)g(used)h(that)i(~)-26 b Fo(g)20 b Fp(solv)o(es)d(\(12)p (#equation.12) [[277 302 288 314] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)26 b(Com)o(bining)17 b(\(17)p (#equation.17) [[367 302 379 314] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (and)g(\(20)p (#equation.20) [[415 302 427 314] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (one)e(arriv)o(es)129 1790 y(at)f(the)g(form)o(ula)646 1911 y Fo(\015)671 1918 y Fk(k)706 1911 y Fp(=)e Fo(\013)789 1918 y Fk(k)q Fs(+2)867 1911 y Fn(\000)916 1841 y Fl(\022)953 1911 y Fo(k)f Fp(+)1045 1877 y(7)p 1045 1899 25 2 v 1045 1945 a(2)1075 1841 y Fl(\023)1119 1911 y Fo(\013)1150 1918 y Fk(k)q Fs(+1)1217 1911 y Fo(:)430 b Fp(\(28\))129 2044 y(No)o(w)18 b(w)o(e)h(can)g(compare)f(the)g(functions)h Fo(g)r Fp(\()p Fo(z)r Fp(\))g(and)g Fo(')p Fp(\()p Fo(z)r Fp(\).)29 b(Let)19 b(us)g(c)o(ho)q(ose)h Fo(\025)e(>)h Fp(1)g(in)129 2104 y(\(21)p (#equation.21) [[108 212 119 224] [1 1 1 [3 3]] [0 0 1]] pdfm (\))e (resp.)24 b(\(22)p (#equation.22) [[162 212 174 224] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)g (Supp)q(ose)18 b(that)g Fo(')p Fp(\()p Fo(z)r Fp(\))d(=)g Fo(g)r Fp(\()p Fo(z)r Fp(\))i(at)g(some)f(p)q(oin)o(t)i Fo(z)r Fp(.)23 b(Then,)17 b(o)o(wing)g(to)129 2164 y(\(12)p (#equation.12) [[108 198 119 210] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (and)f(\(24)p (#equation.24) [[155 198 166 210] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)h (it)f(holds)466 2290 y Fo(')498 2270 y Fr(0)510 2290 y Fp(\()p Fo(z)r Fp(\))11 b Fn(\000)g Fo(g)659 2270 y Fr(0)671 2290 y Fp(\()p Fo(z)r Fp(\))i(=)872 2257 y(1)p 804 2279 162 2 v 804 2325 a Fo(\025z)857 2310 y Fs(2)877 2325 y Fo(g)r Fp(\()p Fo(z)r Fp(\))970 2290 y(\()p Fo(g)r Fp(\()p Fo(z)r Fp(\))e Fn(\000)g Fp(1\)\()p Fo(g)r Fp(\()p Fo(z)r Fp(\))g Fn(\000)g Fo(\025)p Fp(\))p Fo(;)129 2423 y Fp(when)16 b Fo(n)e Fp(=)g(0,)i(and)h(using)f(\(25)p (#equation.25) [[227 135 239 147] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)h (\(23)p (#equation.23) [[255 135 267 147] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g(and)g(\(28)p (#equation.28) [[303 135 314 147] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)415 2572 y Fo(')447 2551 y Fr(0)458 2572 y Fp(\()p Fo(z)r Fp(\))11 b Fn(\000)g Fo(g)607 2551 y Fr(0)619 2572 y Fp(\()p Fo(z)r Fp(\))41 b(=)862 2538 y(1)p 808 2560 133 2 v 808 2606 a Fo(z)833 2592 y Fs(2)852 2606 y Fo(g)r Fp(\()p Fo(z)r Fp(\))945 2572 y(\()p Fo(g)r Fp(\()p Fo(z)r Fp(\))11 b Fn(\000)g Fp(1\))1156 2551 y Fs(2)1187 2572 y Fn(\000)1240 2509 y Fk(n)p Fs(+1)1237 2524 y Fl(X)1241 2631 y Fk(k)q Fs(=2)1317 2572 y Fo(')1349 2579 y Fk(k)1370 2572 y Fo(z)1395 2551 y Fk(k)q Fr(\000)p Fs(2)723 2694 y Fp(=)42 b(\(1)11 b Fn(\000)g Fo(\025)p Fp(\))p Fo(\013)985 2701 y Fk(n)p Fs(+1)1054 2694 y Fo(z)1079 2673 y Fk(n)p Fr(\000)p Fs(1)1158 2694 y Fp(+)g Fo(o)p Fp(\()p Fo(z)1274 2673 y Fk(n)p Fr(\000)p Fs(1)1343 2694 y Fp(\))p Fo(;)914 2819 y Fp(15)p eop %%Page: 16 16 16 15 bop 129 286 a Fp(when)23 b Fo(n)k(>)f Fp(0.)43 b(In)23 b(an)o(y)h(case,)h(there)d(exists)h Fo(\016)28 b(>)e Fp(0)e(s.t.)43 b Fo(')p Fp(\()p Fo(z)r Fp(\))26 b(=)g Fo(g)r Fp(\()p Fo(z)r Fp(\))d(implies)129 347 y Fo(')161 329 y Fr(0)172 347 y Fp(\()p Fo(z)r Fp(\))16 b Fo(<)g(g)330 329 y Fr(0)342 347 y Fp(\()p Fo(z)r Fp(\),)h Fn(8)p Fo(z)g Fn(2)8 b Fp(]0)p Fo(;)g(\016)r Fp([)17 b(\(in)g(the)g(case)h Fo(n)e Fp(=)g(0,)i(w)o(e)f(need)g(also)h(that)g Fo(g)r Fp(\()p Fo(z)r Fp(\))e Fo(>)g Fp(1)i(if)f Fo(z)129 407 y Fp(is)h(su\016cien)o(tly)e(close)i(to)h(0,)g(see)f(Prop)q (osition)h(6)p (#thm.6) [[314 619 320 631] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)29 b(This)18 b(means)g(that)g(the)h(functions)129 467 y Fo(g)r Fp(\()p Fo(z)r Fp(\))c(and)h Fo(')p Fp(\()p Fo(z)r Fp(\))g(coincide)e(in)h(at)h(most)f(one)h(p)q(oin)o(t)g Fo(z)f Fn(2)8 b Fp(]0)p Fo(;)g(\016)r Fp([.)21 b(F)l(urthermore,)13 b(w)o(e)i(ha)o(v)o(e)129 527 y(already)h(sho)o(wn)h(that)f Fo(a)580 534 y Fr(\000)623 527 y Fn(\024)e Fo(\013)707 534 y Fk(n)p Fs(+1)776 527 y Fp(,)h(and)i(so,)g(using)f(also)h(\(26)p (#equation.26) [[364 591 376 603] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)g (w)o(e)e(conclude)h(that)373 642 y(lim)8 b(inf)414 673 y Fk(z)q Fr(#)p Fs(0)521 608 y Fo(g)r Fp(\()p Fo(z)r Fp(\))j Fn(\000)g Fo(')p Fp(\()p Fo(z)r Fp(\))p 521 630 244 2 v 596 676 a Fo(z)621 661 y Fk(n)p Fs(+1)784 642 y Fp(=)i Fo(a)861 649 y Fr(\000)902 642 y Fn(\000)d Fo(\025)e(a)1013 649 y Fk(n)p Fs(+1)1096 642 y Fn(\024)14 b Fp(\(1)d Fn(\000)g Fo(\025)p Fp(\))p Fo(\013)1331 649 y Fk(n)p Fs(+1)1414 642 y Fo(<)j Fp(0)p Fo(:)129 753 y Fp(Th)o(us)k(there)g(exists)g(a)g (sequence)f Fn(f)p Fo(z)812 760 y Fk(n)835 753 y Fn(g)i Fp(s.t.)27 b Fo(z)995 760 y Fk(n)1035 753 y Fn(#)17 b Fp(0)i(and)g Fo(g)r Fp(\()p Fo(z)1284 760 y Fk(n)1307 753 y Fp(\))e Fo(<)g(')p Fp(\()p Fo(z)1472 760 y Fk(n)1495 753 y Fp(\),)i Fn(8)p Fo(n)p Fp(.)26 b(Con-)129 813 y(sequen)o(tly)l(,) 18 b Fo(g)r Fp(\()p Fo(z)r Fp(\))i Fo(<)f(')p Fp(\()p Fo(z)r Fp(\))h(on)g(a)g(righ)o(t)f(neigh)o(b)q(ourho)q(o)q(d)j(of)e(0.) 32 b(Hence,)18 b(in)i(virtue)e(of)129 873 y(\(26)p (#equation.26) [[108 507 119 519] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)495 991 y Fo(a)521 998 y Fs(+)564 991 y Fn(\024)c Fp(lim)8 b(sup)665 1033 y Fk(z)q Fr(#)p Fs(0)779 958 y Fo(')p Fp(\()p Fo(z)r Fp(\))j Fn(\000)935 920 y Fl(P)988 933 y Fk(n)988 972 y(k)q Fs(=0)1062 958 y Fo(\013)1093 965 y Fk(k)1115 958 y Fo(z)1140 939 y Fk(k)p 779 980 382 2 v 924 1026 a Fo(z)949 1011 y Fk(k)q Fs(+1)1180 991 y Fp(=)i Fo(\025)8 b(\013)1298 998 y Fk(n)p Fs(+1)1368 991 y Fo(:)129 1109 y Fp(The)16 b(claim)e(is)i(a)h(consequence)e(of)i (the)f(limit)d Fo(\025)h Fn(#)g Fp(1.)p 1712 1109 2 33 v 1714 1078 30 2 v 1714 1109 V 1743 1109 2 33 v 129 1222 a Fh(Corollary)k(11.)24 b Fg(The)18 b(left)h(maximal)f(solution)h Fo(g)r Fp(\()p Fo(z)r Fp(\))p Fg(,)g(after)f(having)h(b)n(e)n(en)f (de\014ne)n(d)i(at)129 1282 y Fo(z)15 b Fp(=)f(0)k Fg(by)g Fo(g)r Fp(\(0\))c(=)g(1)p Fg(,)j(b)n(elongs)i(to)f Fo(C)799 1264 y Fr(1)836 1282 y Fp(\([)8 b(0)p Fo(;)g(z)946 1289 y Fs(0)973 1282 y Fp(]\))p Fg(.)129 1364 y(Pr)n(o)n(of.)22 b Fp(Observ)o(e)15 b(that)h(consecutiv)o(e)e(di\013eren)o(tiation)h(of) h(equation)g(\(12)p (#equation.12) [[412 390 424 402] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (join)o(tly)f(with)129 1424 y(Prop)q(osition)j(10)p (#thm.10) [[166 375 178 387] [1 1 1 [3 3]] [0 0 1]] pdfm 18 w(imply)d(that,)j(for)f(an)o(y)h Fo(m)d Fn(2)h Fm(Z)1029 1431 y Fs(+)1056 1424 y Fp(,)h Fo(z)1112 1406 y Fk(m)p Fs(+1)1190 1424 y Fo(g)1215 1406 y Fs(\()p Fk(m)p Fs(\))1276 1424 y Fp(\()p Fo(z)r Fp(\))g(has)h(an)g(asymptotic)129 1485 y(series)13 b(at)h Fo(z)h Fp(=)f(0)g(whic)o(h)f(w)o(e)g(shall)h (call)847 1447 y Fl(P)899 1460 y Fr(1)899 1499 y Fk(k)q Fs(=0)974 1485 y Fo(\013)1005 1467 y Fk(m)1005 1498 y(k)1039 1485 y Fo(z)1064 1467 y Fk(k)1093 1485 y Fo(:)p Fp(W)l(e)f(ha)o(v)o(e)g (to)h(sho)o(w)g(that)g Fo(g)i Fn(2)e Fo(C)1701 1467 y Fk(m)1734 1485 y Fp(,)129 1545 y Fn(8)p Fo(m)p Fp(,)f(and)j(this)f(in)f (turn)i(amoun)o(ts)e(to)i(sho)o(wing)f(that)h Fo(\013)1153 1527 y Fk(m)1153 1558 y(k)1200 1545 y Fp(=)e(0)i(for)f Fo(k)h(<)e(m)8 b Fp(+)h(1.)21 b(Let)15 b(us)129 1605 y(pro)q(ceed)i(b)o(y)f(induction)h(in)g Fo(m)p Fp(.)23 b(The)17 b(case)g Fo(m)e Fp(=)g(0)i(w)o(as)h(the)f(con)o(ten)o(t)f(of)h (Prop)q(osition)129 1665 y(6)p (#thm.6) [[103 317 109 329] [1 1 1 [3 3]] [0 0 1]] pdfm (.)28 b(Assume)17 b(no)o(w)i(that)g Fo(g)h Fn(2)f Fo(C)724 1647 y Fk(m)756 1665 y Fp(.)29 b(Then)18 b Fo(\013)959 1647 y Fk(m)959 1678 y(k)1011 1665 y Fp(=)g(0)h(for)f Fo(k)i(<)e(m)12 b Fp(+)h(1,)19 b(and)g(the)g(mean)129 1725 y(v)m(alue)d(theorem)e(implies)g(that)322 1843 y(lim)8 b(inf)363 1874 y Fk(z)q Fr(#)p Fs(0)466 1843 y Fo(g)491 1822 y Fs(\()p Fk(m)p Fs(+1\))597 1843 y Fp(\()p Fo(z)r Fp(\))13 b Fn(\024)731 1809 y Fo(dg)781 1791 y Fs(\()p Fk(m)p Fs(\))842 1809 y Fp(\(0)885 1816 y Fs(+)915 1809 y Fp(\))p 731 1831 204 2 v 807 1877 a Fo(dz)953 1843 y Fp(=)h Fo(\013)1036 1822 y Fk(m)1036 1855 y(m)p Fs(+2)1128 1843 y Fn(\024)f Fp(lim)8 b(sup)1229 1884 y Fk(z)q Fr(#)p Fs(0)1338 1843 y Fo(g)1363 1822 y Fs(\()p Fk(m)p Fs(+1\))1469 1843 y Fp(\()p Fo(z)r Fp(\))g Fo(:)107 b Fp(\(29\))129 1969 y(On)19 b(the)h(other)f(hand,)i(since)e Fo(z)719 1951 y Fk(m)p Fs(+2)797 1969 y Fo(g)822 1951 y Fs(\()p Fk(m)p Fs(+1\))928 1969 y Fp(\()p Fo(z)r Fp(\))g(has)i(an)f(asymptotic) e(series)h(the)g(limit)129 2029 y(lim)196 2036 y Fk(z)q Fr(#)p Fs(0)260 2029 y Fo(g)285 2011 y Fs(\()p Fk(m)p Fs(+1\))391 2029 y Fp(\()p Fo(z)r Fp(\))g(alw)o(a)o(ys)g(exists)g(and)g (equals)g(either)f Fn(\0061)h Fp(or)h Fo(\013)1361 2009 y Fk(m)p Fs(+1)1361 2041 y Fk(m)p Fs(+2)1459 2029 y Fp(dep)q(ending)f (on)129 2089 y(whether)d(there)g(exists)g(an)h(index)e Fo(k)h(<)f(m)10 b Fp(+)i(2)17 b(s.t.)k Fo(\013)1131 2069 y Fk(m)p Fs(+1)1131 2103 y Fk(k)1224 2089 y Fn(6)p Fp(=)14 b(0)j(or)g(not.)22 b(Ho)o(w)o(ev)o(er)15 b(the)129 2150 y(prop)q(ert)o(y)h(\(29)p (#equation.29) [[155 201 167 213] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (clearly)e(excludes)g(the)h(\014rst)g(p)q(ossibilit)o(y)l(.)p 1712 2150 2 33 v 1714 2118 30 2 v 1714 2150 V 1743 2150 2 33 v 129 2313 a Fq(4)81 b(Asymptotics)20 b(of)h(a)g(solution)f Fd(h)p Fc(\()p Fd(t)p Fc(\))i Fq(of)f(the)g(second)250 2404 y(order)27 b(di\013eren)n(tial)d(equation)129 2513 y Fp(Except)11 b(of)h(the)g(last)g(subsection,)g(w)o(e)g(still)f (consider)g(the)h(particular)f(case)h(when)g Fo(t)1638 2520 y Fs(0)1671 2513 y Fp(=)i(0)129 2574 y(and)20 b Fo(h)255 2581 y Fs(1)293 2574 y Fo(>)f Fp(0)h(\(see)f(the)g(remark)f (at)h(the)g(end)h(of)f(Section)g(2)p (#section.2) [[367 99 373 111] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)31 b(W)l(e)19 b(shall)g(pro)q(ceed)g(to)129 2634 y(the)i(case)h(of)h (general)f(initial)e(condition)i(only)g(at)g(the)g(v)o(ery)f(end)h(of)g (the)g(pro)q(of,)i(in)129 2694 y(Subsection)16 b(4.5)p (#subsection.4.5) [[162 70 176 82] [1 1 1 [3 3]] [0 0 1]] pdfm (.)914 2819 y(16)p eop %%Page: 17 17 17 16 bop 129 286 a Ff(4.1)66 b(Reduction)17 b(of)e(the)i(second)e (order)i(di\013eren)n(tial)i(equation)129 454 y Fp(Let)k(us)g(no)o(w)h (complete)c(some)i(computations)h(concerning)f(the)h(reduction)g(of)g (the)129 514 y(second)12 b(order)g(di\013eren)o(tial)e(equation)i(\(1)p (#equation.1) [[278 594 284 606] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)g (\(2)p (#equation.2) [[299 594 305 606] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g(to)h(a)f(\014rst)g(order)g(di\013eren)o(tial)e(equation.)129 574 y(Let)16 b Fo(g)r Fp(\()p Fo(z)r Fp(\))g(b)q(e)h(the)f(left)f (maximal)e(solution)k(of)f(the)g(\014rst)h(order)f(di\013eren)o(tial)f (equation)453 641 y Fl(\022)489 712 y Fp(1)d Fn(\000)579 678 y Fp(3)p 579 700 25 2 v 579 746 a(4)609 712 y Fo(z)e(g)r Fp(\()p Fo(z)r Fp(\))h Fn(\000)g Fo(z)816 691 y Fs(2)835 712 y Fo(g)860 691 y Fr(0)872 712 y Fp(\()p Fo(z)r Fp(\))935 641 y Fl(\023)980 712 y Fo(g)r Fp(\()p Fo(z)r Fp(\))j(=)g(1)p Fo(;)24 b(g)r Fp(\()p Fo(z)1263 719 y Fs(0)1283 712 y Fp(\))13 b(=)h Fo(g)1390 719 y Fs(0)1410 712 y Fo(;)129 846 y Fp(where)723 965 y Fo(z)746 972 y Fs(0)779 965 y Fp(=)848 932 y(4)p 836 954 48 2 v 836 999 a Fo(h)864 982 y Fs(4)864 1012 y(0)889 965 y Fo(;)24 b(g)950 972 y Fs(0)984 965 y Fp(=)14 b Fo(h)1064 945 y Fs(3)1064 978 y(0)1083 965 y Fo(h)1111 972 y Fs(1)1140 965 y Fo(:)129 1102 y Fp(>F)l(rom)22 b(Section)g(3)p (#section.3) [[183 453 189 465] [1 1 1 [3 3]] [0 0 1]] pdfm 24 w(w)o(e)g(kno)o(w)h(that)h Fo(g)r Fp(\()p Fo(z)r Fp(\))f(is)g(a)g(p) q(ositiv)o(e)g(function)g(from)f(the)h(class)129 1162 y Fo(C)168 1144 y Fr(1)205 1162 y Fp(\([)8 b(0)p Fo(;)g(z)315 1169 y Fs(0)342 1162 y Fp(]\))16 b(\(Prop)q(osition)h(6)p (#thm.6) [[233 438 239 450] [1 1 1 [3 3]] [0 0 1]] pdfm 17 w(and)g(Corollary)f(11)p (#thm.11) [[318 438 329 450] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)22 b(Consider)17 b(the)f(function)596 1296 y Fo(G)p Fp(\()p Fo(x)p Fp(\))e(=)766 1229 y Fl(Z)816 1242 y Fk(x)794 1341 y(h)814 1330 y Fj(4)814 1352 y(0)880 1263 y Fo(ds)p 851 1285 107 2 v 851 1333 a(g)884 1293 y Fl(\000)912 1314 y Fs(4)p 912 1322 18 2 v 913 1351 a Fk(s)935 1293 y Fl(\001)971 1296 y Fo(;)24 b(h)1037 1276 y Fs(4)1037 1309 y(0)1071 1296 y Fn(\024)13 b Fo(x)h(<)g Fn(1)p Fo(:)380 b Fp(\(30\))129 1445 y(Then)18 b Fo(G)g Fn(2)g Fo(C)404 1426 y Fr(1)441 1445 y Fp(\([)8 b Fo(h)510 1426 y Fs(4)510 1457 y(0)530 1445 y Fo(;)g Fn(1)p Fp([)g(\),)18 b Fo(G)h Fp(is)f(strictly)f(increasing,)i Fo(G)p Fp(\()p Fo(h)1282 1426 y Fs(4)1282 1457 y(0)1302 1445 y Fp(\))f(=)f(0,)i(and,)g(o)o(wing) g(to)129 1505 y(\(14)p (#equation.14) [[108 356 119 368] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)d (lim)313 1512 y Fk(x)p Fr(!1)414 1505 y Fo(G)p Fp(\()p Fo(x)p Fp(\))e(=)g Fn(1)p Fp(.)22 b(So)17 b(the)f(in)o(v)o(erse)f (function)h(satis\014es)h Fo(G)1393 1487 y Fr(\000)p Fs(1)1455 1505 y Fn(2)d Fo(C)1541 1487 y Fr(1)1578 1505 y Fp(\([)8 b(0)p Fo(;)g Fn(1)p Fp([\))129 1565 y(with)16 b Fo(G)278 1547 y Fr(\000)p Fs(1)325 1565 y Fp(\(0\))f(=)e Fo(h)481 1547 y Fs(4)481 1577 y(0)501 1565 y Fp(.)21 b(Set)715 1684 y Fo(h)p Fp(\()p Fo(t)p Fp(\))14 b(=)865 1644 y Fl(\000)887 1684 y Fo(G)925 1664 y Fr(\000)p Fs(1)981 1684 y Fp(\(4)p Fo(t)p Fp(\))1061 1644 y Fl(\001)1084 1655 y Fs(1)p Fk(=)p Fs(4)1147 1684 y Fo(:)129 1793 y Fp(Then)i Fo(h)p Fp(\()p Fo(t)p Fp(\))g(solv)o(es)g(the)g(problem)e (\(1)p (#equation.1) [[261 287 267 299] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)i(\(2)p (#equation.2) [[283 287 289 299] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)202 1854 y(Actually)l(,)e Fo(G)p Fp(\()p Fo(h)p Fp(\()p Fo(t)p Fp(\))550 1836 y Fs(4)570 1854 y Fp(\))g(=)f(4)p Fo(t)p Fp(,)j Fo(G)764 1836 y Fr(0)776 1854 y Fp(\()p Fo(h)823 1836 y Fs(4)843 1854 y Fp(\))e(=)g Fo(g)r Fp(\(4)p Fo(=h)1048 1836 y Fs(4)1068 1854 y Fp(\))1087 1836 y Fr(\000)p Fs(1)1135 1854 y Fp(,)h(and)i(so)562 1992 y Fo(h)590 1972 y Fs(3)610 1992 y Fo(h)638 1972 y Fr(0)664 1992 y Fp(=)788 1959 y(1)p 721 1981 161 2 v 721 2026 a(4)p Fo(G)783 2012 y Fr(0)795 2026 y Fp(\()p Fo(h)842 2012 y Fs(4)862 2026 y Fp(\))891 1959 y Fo(d)8 b(G)p Fp(\()p Fo(h)1009 1940 y Fs(4)1030 1959 y Fp(\))p 891 1981 158 2 v 948 2026 a Fo(dt)1067 1992 y Fp(=)14 b Fo(g)1153 1922 y Fl(\022)1206 1959 y Fp(4)p 1194 1981 48 2 v 1194 2026 a Fo(h)1222 2012 y Fs(4)1247 1922 y Fl(\023)1300 1992 y Fo(:)347 b Fp(\(31\))129 2130 y(Di\013eren)o(tiating)15 b(\(31)p (#equation.31) [[185 206 197 218] [1 1 1 [3 3]] [0 0 1]] pdfm (\))i (once)f(more)f(giv)o(es)533 2268 y(3)p Fo(h)585 2247 y Fs(2)606 2268 y Fp(\()p Fo(h)653 2247 y Fr(0)664 2268 y Fp(\))683 2247 y Fs(2)714 2268 y Fp(+)c Fo(h)791 2247 y Fs(3)811 2268 y Fo(h)839 2247 y Fr(0)o(0)874 2268 y Fp(=)j Fn(\000)p Fp(16)8 b Fo(h)1049 2247 y Fr(\000)p Fs(5)1097 2268 y Fo(h)1125 2247 y Fr(0)1137 2268 y Fo(g)1162 2247 y Fr(0)1182 2197 y Fl(\022)1235 2234 y Fp(4)p 1223 2256 V 1223 2302 a Fo(h)1251 2287 y Fs(4)1276 2197 y Fl(\023)1329 2268 y Fo(:)129 2407 y Fp(Denote)16 b(for)h(brevit)o(y)d Fo(z)i Fp(=)e(4)p Fo(=h)704 2389 y Fs(4)724 2407 y Fp(.)21 b(Hence)354 2516 y Fo(h)382 2496 y Fs(3)402 2516 y Fp(\()p Fo(h)449 2496 y Fr(0)472 2516 y Fp(+)11 b Fo(h)549 2496 y Fr(00)570 2516 y Fp(\))42 b(=)f Fo(g)r Fp(\()p Fo(z)r Fp(\))11 b Fn(\000)g Fp(3)p Fo(h)911 2496 y Fr(\000)p Fs(4)959 2516 y Fp(\()p Fo(h)1006 2496 y Fs(3)1025 2516 y Fo(h)1053 2496 y Fr(0)1065 2516 y Fp(\))1084 2496 y Fs(2)1115 2516 y Fn(\000)g Fp(16)d Fo(h)1249 2496 y Fr(\000)p Fs(8)1297 2516 y Fp(\()p Fo(h)1344 2496 y Fs(3)1364 2516 y Fo(h)1392 2496 y Fr(0)1403 2516 y Fp(\))p Fo(g)1447 2496 y Fr(0)1459 2516 y Fp(\()p Fo(z)r Fp(\))631 2611 y(=)41 b Fo(g)r Fp(\()p Fo(z)r Fp(\))11 b Fn(\000)864 2577 y Fp(3)p 864 2600 25 2 v 864 2645 a(4)893 2611 y Fo(z)f(g)r Fp(\()p Fo(z)r Fp(\))1014 2591 y Fs(2)1045 2611 y Fn(\000)h Fo(z)1120 2591 y Fs(2)1140 2611 y Fo(g)r Fp(\()p Fo(z)r Fp(\))d Fo(g)1261 2591 y Fr(0)1273 2611 y Fp(\()p Fo(z)r Fp(\))631 2694 y(=)41 b(1)8 b Fo(:)914 2819 y Fp(17)p eop %%Page: 18 18 18 17 bop 129 288 a Fp(F)l(urthermore,)14 b Fo(h)p Fp(\(0\))g(=)g(\()p Fo(G)638 270 y Fr(\000)p Fs(1)685 288 y Fp(\(0\)\))767 262 y Fs(1)p Fk(=)p Fs(4)835 288 y Fp(=)g Fo(h)915 295 y Fs(0)951 288 y Fp(and)359 435 y Fo(h)387 414 y Fr(0)399 435 y Fp(\(0\))42 b(=)f Fo(G)620 414 y Fr(\000)p Fs(1)668 435 y Fp(\(0\))730 414 y Fr(\000)p Fs(3)p Fk(=)p Fs(4)837 401 y Fo(d)p 826 423 49 2 v 826 469 a(ds)879 435 y(G)917 414 y Fr(\000)p Fs(1)965 435 y Fp(\(0\))22 b(=)g Fo(h)1137 414 y Fr(\000)p Fs(3)1137 447 y(0)1201 365 y Fl(\022)1257 401 y Fo(d)p 1243 423 54 2 v 1243 469 a(dx)1301 435 y(G)p Fp(\()p Fo(h)1386 414 y Fs(4)1386 447 y(0)1406 435 y Fp(\))1425 365 y Fl(\023)1462 376 y Fr(\000)p Fs(1)503 571 y Fp(=)41 b Fo(h)610 550 y Fr(\000)p Fs(3)610 583 y(0)666 571 y Fo(g)699 501 y Fl(\022)752 537 y Fp(4)p 741 559 48 2 v 741 605 a Fo(h)769 588 y Fs(4)769 617 y(0)794 501 y Fl(\023)852 571 y Fp(=)22 b Fo(h)940 578 y Fs(1)969 571 y Fo(:)129 737 y Ff(4.2)66 b(Asymptotics)22 b(of)g Fe(G)p Fz(\()p Fe(x)p Fz(\))129 829 y Fp(First)16 b(let)h(us)g(\014nd,)h(in)f Fm(C)9 b Fp([[)p Fo(z)r Fp(]],)18 b(the)f(recipro)q(cal)f(elemen)o(t)e(to)k(the)f(formal)f(p)q(o)o(w)o (er)h(series)131 889 y(~)-26 b Fo(g)r Fp(\()p Fo(z)r Fp(\))13 b(=)282 852 y Fl(P)335 865 y Fr(1)335 904 y Fk(k)q Fs(=0)410 889 y Fo(\013)441 896 y Fk(k)462 889 y Fo(z)487 871 y Fk(k)524 889 y Fp(de\014ned)j(in)g(\(17)p (#equation.17) [[256 504 268 516] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)h (\(18)p (#equation.18) [[284 504 296 516] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)22 b(Set)739 1040 y(~)-26 b Fo(g)r Fp(\()p Fo(z)r Fp(\))825 1019 y Fr(\000)p Fs(1)886 1040 y Fp(=)956 977 y Fr(1)938 992 y Fl(X)940 1097 y Fk(n)p Fs(=0)1018 1040 y Fo(\014)1046 1047 y Fk(n)1069 1040 y Fo(z)1094 1019 y Fk(n)1126 1040 y Fo(:)129 1193 y Fp(The)17 b(formal)f(p)q(o)o(w)o(er) i(series)g(~)-26 b Fo(g)s Fp(\()p Fo(z)r Fp(\))17 b(solv)o(es)g(the)g (di\013eren)o(tial)f(equation)h(\(12)p (#equation.12) [[421 431 432 443] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (and)g(so)g(an)129 1253 y(easy)e(calculation)g(sho)o(ws)h(that)739 1240 y(~)728 1253 y Fo(f)6 b Fp(\()p Fo(z)r Fp(\))13 b(=)j(~)-26 b Fo(g)r Fp(\()p Fo(z)r Fp(\))974 1235 y Fr(\000)p Fs(1)1038 1253 y Fp(solv)o(es)15 b(the)h(di\013eren)o(tial)f (equation)530 1391 y Fo(f)559 1370 y Fr(0)571 1391 y Fp(\()p Fo(z)r Fp(\))e(=)h Fn(\000)753 1357 y Fp(1)p 743 1379 45 2 v 743 1425 a Fo(z)768 1411 y Fs(2)801 1391 y Fo(f)5 b Fp(\()p Fo(z)r Fp(\))893 1370 y Fs(2)913 1391 y Fp(\(1)11 b Fn(\000)g Fo(f)5 b Fp(\()p Fo(z)r Fp(\)\))11 b(+)1193 1357 y(3)p 1193 1379 25 2 v 1193 1425 a(4)1228 1357 y Fo(f)5 b Fp(\()p Fo(z)r Fp(\))p 1228 1379 93 2 v 1261 1425 a Fo(z)1333 1391 y(:)129 1514 y Fp(On)17 b(the)g(other)g(hand,)g(this)g(di\013eren)o(tial)f(equation)h(implies)d (a)k(recursiv)o(e)d(rule)h(on)i(the)129 1574 y(co)q(e\016cien)o(ts)d Fo(\014)404 1581 y Fk(n)427 1574 y Fp(,)g(namely)g(rule)g(\(6)p (#equation.6) [[250 339 256 351] [1 1 1 [3 3]] [0 0 1]] pdfm (\))i (preceding)e(the)h(form)o(ulation)f(of)i(Theorem)e(1)p (#thm.1) [[463 339 469 351] [1 1 1 [3 3]] [0 0 1]] pdfm (.)129 1675 y Fh(Lemm)o(a)h(12.)24 b Fg(The)c(asymptotic)f(series)g(at)g (in\014nity)h(of)f(the)h(function)g Fo(G)p Fp(\()p Fo(x)p Fp(\))g Fg(de\014ne)n(d)129 1736 y(in)d(\(30)p (#equation.30) [[122 300 134 312] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (is)f(given)i(by)485 1884 y Fo(G)p Fp(\()p Fo(x)p Fp(\))14 b Fn(\030)g Fo(x)d Fn(\000)f Fp(3)17 b(ln\()p Fo(x)p Fp(\))11 b(+)g Fo(c)g Fn(\000)g Fp(4)1085 1822 y Fr(1)1066 1837 y Fl(X)1070 1943 y Fk(k)q Fs(=1)1152 1850 y Fo(\014)1180 1857 y Fk(k)q Fs(+1)p 1152 1873 94 2 v 1185 1918 a Fo(k)1259 1814 y Fl(\022)1302 1850 y Fp(4)p 1300 1873 28 2 v 1300 1918 a Fo(x)1333 1814 y Fl(\023)1370 1825 y Fk(k)129 2036 y Fg(wher)n(e)450 2178 y Fo(c)j Fp(=)537 2110 y Fl(Z)587 2123 y Fr(1)565 2223 y Fk(h)590 2211 y Fj(4)585 2233 y(0)633 2092 y Fl( )718 2144 y Fp(1)p 677 2166 107 2 v 677 2214 a Fo(g)710 2174 y Fl(\000)738 2195 y Fs(4)p 738 2203 18 2 v 739 2232 a Fk(s)761 2174 y Fl(\001)800 2178 y Fn(\000)d Fp(1)g(+)939 2144 y(3)p 939 2166 25 2 v 940 2212 a Fo(s)968 2092 y Fl(!)1016 2178 y Fo(ds)g Fn(\000)g Fo(h)1159 2157 y Fs(4)1153 2190 y(0)1190 2178 y Fp(+)g(3)17 b(ln\()p Fo(h)1374 2157 y Fs(4)1368 2190 y(0)1393 2178 y Fp(\))p Fo(:)235 b Fp(\(32\))129 2328 y Fg(Pr)n(o)n(of.)22 b Fp(It)16 b(holds)367 2479 y Fo(G)p Fp(\()p Fo(x)p Fp(\))41 b(=)592 2411 y Fl(Z)642 2425 y Fr(1)619 2524 y Fk(h)644 2513 y Fj(4)639 2535 y(0)687 2394 y Fl( )773 2446 y Fp(1)p 732 2468 107 2 v 732 2516 a Fo(g)765 2476 y Fl(\000)793 2496 y Fs(4)p 793 2505 18 2 v 794 2533 a Fk(s)815 2476 y Fl(\001)854 2479 y Fn(\000)11 b Fp(1)h(+)994 2446 y(3)p 994 2468 25 2 v 995 2513 a Fo(s)1023 2394 y Fl(!)1071 2479 y Fo(ds)f Fp(+)1179 2411 y Fl(Z)1229 2425 y Fk(x)1207 2524 y(h)1232 2513 y Fj(4)1227 2535 y(0)1260 2409 y Fl(\022)1297 2479 y Fp(1)g Fn(\000)1387 2446 y Fp(3)p 1387 2468 V 1388 2513 a Fo(s)1416 2409 y Fl(\023)1461 2479 y Fo(ds)592 2645 y Fn(\000)639 2577 y Fl(Z)689 2591 y Fr(1)666 2690 y Fk(x)734 2560 y Fl( )820 2612 y Fp(1)p 779 2634 107 2 v 779 2682 a Fo(g)812 2642 y Fl(\000)840 2663 y Fs(4)p 840 2671 18 2 v 841 2699 a Fk(s)863 2642 y Fl(\001)901 2645 y Fn(\000)g Fp(1)h(+)1041 2612 y(3)p 1041 2634 25 2 v 1042 2679 a Fo(s)1070 2560 y Fl(!)1118 2645 y Fo(ds)c(:)914 2819 y Fp(18)p eop %%Page: 19 19 19 18 bop 129 286 a Fp(According)15 b(to)i(Prop)q(osition)g(10)p (#thm.10) [[236 648 247 660] [1 1 1 [3 3]] [0 0 1]] pdfm 17 w(w)o(e)f(ha)o(v)o(e)g(the)g(asymptotics)f(at)h(in\014nit)o(y)l(,) 655 370 y(1)p 613 392 107 2 v 613 440 a Fo(g)647 400 y Fl(\000)675 421 y Fs(4)p 675 429 18 2 v 676 457 a Fk(s)697 400 y Fl(\001)736 403 y Fn(\000)11 b Fp(1)g(+)875 370 y(3)p 875 392 25 2 v 876 438 a Fo(s)919 403 y Fn(\030)990 341 y Fr(1)971 356 y Fl(X)975 462 y Fk(k)q Fs(=2)1051 403 y Fo(\014)1079 410 y Fk(k)1109 333 y Fl(\022)1150 370 y Fp(4)p 1150 392 V 1151 438 a Fo(s)1180 333 y Fl(\023)1216 344 y Fk(k)1254 403 y Fo(:)129 527 y Fp(The)16 b(claim)e(then)i(follo)o (ws)g(straigh)o(tforw)o(ardly)l(.)p 1712 527 2 33 v 1714 495 30 2 v 1714 527 V 1743 527 2 33 v 129 667 a Ff(4.3)66 b(Asymptotics)22 b(of)g Fe(G)794 645 y Fb(\000)p Fx(1)849 667 y Fz(\()p Fe(x)p Fz(\))129 759 y Fp(Let)16 b(us)h(no)o(w)f(fo)q (cus)h(on)g(the)f(in)o(v)o(erse)e(function)i Fo(G)1047 741 y Fr(\000)p Fs(1)1095 759 y Fp(.)129 836 y Fh(Lemm)o(a)g(13.)24 b Fg(Ther)n(e)17 b(exists)i Fo(x)725 843 y Fk(?)761 836 y Fg(such)f(that)g(for)f(al)r(l)h Fo(x)c(>)g(x)1241 843 y Fk(?)1278 836 y Fg(it)j(holds)h(true)g(that)552 926 y Fp(0)c Fn(\024)g Fo(G)681 906 y Fr(\000)p Fs(1)728 926 y Fp(\()p Fo(x)p Fp(\))d Fn(\000)g Fo(x)i Fn(\024)997 892 y Fo(x)p 954 915 113 2 v 954 960 a(x)e Fn(\000)g Fp(4)1080 926 y(\()p Fo(x)g Fn(\000)g Fo(G)p Fp(\()p Fo(x)p Fp(\)\))p Fo(:)336 b Fp(\(33\))129 1029 y Fg(Pr)n(o)n(of.)22 b Fp(Cho)q(ose)c Fo(y)475 1036 y Fk(?)508 1029 y Fn(\025)c Fo(G)p Fp(\()p Fo(h)652 1011 y Fs(4)646 1041 y(0)672 1029 y Fp(\))g Fo(>)g Fp(0)i(so)h(that,)f(for)h(all)f Fo(y)f Fn(\025)f Fo(y)1235 1036 y Fk(?)1254 1029 y Fp(,)523 1132 y(0)g Fn(\024)g Fp(1)d Fn(\000)753 1099 y Fp(1)p 704 1121 122 2 v 704 1184 a Fo(g)738 1129 y Fl(\020)773 1165 y Fs(4)p 772 1173 19 2 v 772 1201 a Fk(y)796 1129 y Fl(\021)845 1132 y Fn(\024)903 1099 y Fp(4)p 902 1121 26 2 v 902 1167 a Fo(y)957 1132 y Fp(and)25 b Fo(y)13 b Fn(\000)e Fo(G)p Fp(\()p Fo(y)r Fp(\))j Fn(\025)f Fp(0)p Fo(:)308 b Fp(\(34\))129 1279 y(This)12 b(is)f(p)q(ossible)h(o)o(wing)h (to)f(Prop)q(osition)h(10)p (#thm.10) [[290 410 301 422] [1 1 1 [3 3]] [0 0 1]] pdfm 12 w(and)g(Lemma)d(12)p (#thm.12) [[367 410 378 422] [1 1 1 [3 3]] [0 0 1]] pdfm (.)20 b(Set)12 b Fo(x)1419 1286 y Fk(?)1452 1279 y Fp(=)i(max)o Fn(f)p Fp(4)p Fo(;)8 b(y)1690 1286 y Fk(?)1709 1279 y Fn(g)p Fp(.)129 1339 y(F)l(or)16 b Fo(x)e(>)f(x)337 1346 y Fk(?)373 1339 y Fp(\014xed)j(de\014ne)g(a)h(sequence)e Fn(f)p Fo(y)924 1346 y Fk(n)947 1339 y Fn(g)972 1321 y Fr(1)972 1351 y Fk(n)p Fs(=0)1057 1339 y Fp(b)o(y)g(the)h(recursiv)o (e)f(rule)605 1420 y Fo(y)629 1427 y Fs(0)662 1420 y Fp(=)f Fo(x;)24 b(y)804 1427 y Fk(n)p Fs(+1)886 1420 y Fp(=)14 b Fo(x)d Fp(+)g Fo(y)1050 1427 y Fk(n)1084 1420 y Fn(\000)g Fo(G)p Fp(\()p Fo(y)1215 1427 y Fk(n)1239 1420 y Fp(\))p Fo(:)129 1501 y Fp(Owing)16 b(to)h(our)f(c)o(hoice,)f Fo(y)612 1508 y Fk(n)649 1501 y Fn(\025)f Fo(x)f Fn(\025)h Fo(x)824 1508 y Fk(?)857 1501 y Fn(\025)g Fo(y)934 1508 y Fk(?)969 1501 y Fp(for)j(all)f Fo(n)p Fp(.)21 b(Using)16 b(\(30)p (#equation.30) [[392 357 403 369] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h(one)f(\014nds)h(that)248 1626 y Fo(y)272 1633 y Fk(n)p Fs(+2)351 1626 y Fn(\000)11 b Fo(y)425 1633 y Fk(n)p Fs(+1)507 1626 y Fp(=)j Fo(y)583 1633 y Fk(n)p Fs(+1)662 1626 y Fn(\000)d Fo(y)736 1633 y Fk(n)771 1626 y Fn(\000)820 1558 y Fl(Z)870 1571 y Fk(y)887 1576 y Fi(n)p Fj(+1)848 1671 y Fk(y)865 1675 y Fi(n)992 1592 y Fo(ds)p 963 1615 107 2 v 963 1663 a(g)996 1623 y Fl(\000)1024 1643 y Fs(4)p 1024 1652 18 2 v 1025 1680 a Fk(s)1047 1623 y Fl(\001)1088 1626 y Fp(=)1140 1558 y Fl(Z)1190 1571 y Fk(y)1207 1576 y Fi(n)p Fj(+1)1168 1671 y Fk(y)1185 1675 y Fi(n)1277 1541 y Fl( )1317 1626 y Fp(1)g Fn(\000)1448 1592 y Fp(1)p 1407 1615 107 2 v 1407 1663 a Fo(g)1441 1623 y Fl(\000)1468 1643 y Fs(4)p 1468 1652 18 2 v 1469 1680 a Fk(s)1491 1623 y Fl(\001)1519 1541 y(!)1567 1626 y Fo(ds:)129 1751 y Fp(In)16 b(virtue)f(of)h(\(34)p (#equation.34) [[169 297 181 309] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)h (the)f(sequence)f(satis\014es)345 1854 y(0)f Fn(\024)f Fo(y)459 1861 y Fk(n)p Fs(+2)539 1854 y Fn(\000)e Fo(y)613 1861 y Fk(n)p Fs(+1)695 1854 y Fn(\024)i Fp(4\(ln\()p Fo(y)874 1861 y Fk(n)p Fs(+1)943 1854 y Fp(\))e Fn(\000)g Fp(ln)o(\()p Fo(y)1106 1861 y Fk(n)1129 1854 y Fp(\)\))j Fn(\024)1240 1820 y Fp(4)p 1239 1843 28 2 v 1239 1888 a Fo(x)1280 1854 y Fp(\()p Fo(y)1323 1861 y Fk(n)p Fs(+1)1402 1854 y Fn(\000)d Fo(y)1476 1861 y Fk(n)1499 1854 y Fp(\))p Fo(:)129 1951 y Fp(Since)k Fo(y)280 1958 y Fs(1)311 1951 y Fn(\000)10 b Fo(y)384 1958 y Fs(0)418 1951 y Fp(=)k Fo(x)c Fn(\000)h Fo(G)p Fp(\()p Fo(x)p Fp(\))17 b(w)o(e)e(get)518 2062 y(0)g Fn(\024)e Fo(y)633 2069 y Fk(n)p Fs(+1)713 2062 y Fn(\000)d Fo(y)786 2069 y Fk(n)823 2062 y Fn(\024)876 1991 y Fl(\022)919 2028 y Fp(4)p 918 2050 V 918 2096 a Fo(x)950 1991 y Fl(\023)987 2003 y Fk(n)1019 2062 y Fp(\()p Fo(x)h Fn(\000)f Fo(G)p Fp(\()p Fo(x)p Fp(\)\))p Fo(;)25 b Fn(8)p Fo(n:)129 2171 y Fp(By)17 b(the)i(c)o(hoice)e(of)h Fo(x)526 2178 y Fk(?)564 2171 y Fp(w)o(e)g(ha)o(v)o(e)g(4)g Fo(<)f(x)i Fp(and,)g(consequen)o(tly)l(,)e(the)h(sequence)f Fn(f)p Fo(y)1648 2178 y Fk(n)1671 2171 y Fn(g)i Fp(is)129 2232 y(con)o(v)o(ergen)o(t.)28 b(The)19 b(limit)e Fo(y)j Fp(=)f(lim)6 b Fo(y)824 2239 y Fk(n)866 2232 y Fp(solv)o(es)19 b(0)g(=)g Fo(x)12 b Fn(\000)h Fo(G)p Fp(\()p Fo(y)r Fp(\))19 b(and)h(so)g Fo(y)g Fp(=)f Fo(G)1621 2214 y Fr(\000)p Fs(1)1668 2232 y Fp(\()p Fo(x)p Fp(\).)129 2292 y(Moreo)o(v)o(er,)227 2408 y(0)30 b Fn(\024)h Fo(y)12 b Fn(\000)f Fo(y)461 2415 y Fk(n)526 2408 y Fp(=)624 2346 y Fr(1)606 2361 y Fl(X)608 2467 y Fk(k)q Fs(=)p Fk(n)678 2408 y Fp(\()p Fo(y)721 2415 y Fk(k)q Fs(+1)798 2408 y Fn(\000)g Fo(y)872 2415 y Fk(k)893 2408 y Fp(\))526 2571 y Fn(\024)624 2508 y Fr(1)606 2523 y Fl(X)608 2629 y Fk(k)q Fs(=)p Fk(n)686 2500 y Fl(\022)730 2537 y Fp(4)p 728 2559 V 728 2605 a Fo(x)761 2500 y Fl(\023)797 2512 y Fk(k)827 2571 y Fp(\()p Fo(x)g Fn(\000)f Fo(G)p Fp(\()p Fo(x)p Fp(\)\))k(=)1171 2537 y Fo(x)p 1128 2559 113 2 v 1128 2605 a(x)d Fn(\000)g Fp(4)1254 2571 y(\()p Fo(x)g Fn(\000)g Fo(G)p Fp(\()p Fo(x)p Fp(\)\))1493 2500 y Fl(\022)1537 2537 y Fp(4)p 1535 2559 28 2 v 1535 2605 a Fo(x)1568 2500 y Fl(\023)1604 2512 y Fk(n)1636 2571 y Fo(:)129 2694 y Fp(The)16 b(particular)g(case)g Fo(n)e Fp(=)g(0)i(in)g(this)g(relation)g(is)g(nothing)h(but)g(our)f (claim.)p 1712 2694 2 33 v 1714 2663 30 2 v 1714 2694 V 1743 2694 2 33 v 914 2819 a(19)p eop %%Page: 20 20 20 19 bop 129 286 a Fp(Com)o(bining)15 b(Lemma)f(13)p (#thm.13) [[204 648 216 660] [1 1 1 [3 3]] [0 0 1]] pdfm 17 w(with)i(Lemma)e(12)p (#thm.12) [[288 648 300 660] [1 1 1 [3 3]] [0 0 1]] pdfm 17 w(one)i(immediatel)o(y)d(gets)129 377 y Fh(Corollary)18 b(14.)24 b Fg(It)17 b(holds)h(true)f(that,)h(as)f Fo(x)d Fn(!)f Fp(+)p Fn(1)p Fg(,)688 475 y Fo(G)726 454 y Fr(\000)p Fs(1)773 475 y Fp(\()p Fo(x)p Fp(\))h(=)f Fo(x)e Fp(+)g Fo(O)q Fp(\(ln)q(\()p Fo(x)p Fp(\)\))p Fo(:)202 572 y Fp(Recall)j(that)h(in)f(\(4)p (#equation.4) [[198 580 204 592] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)i (\(5)p (#equation.5) [[220 580 226 592] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f(w)o(e)f(ha)o(v)o(e)g(in)o(tro)q(duced)h(p)q(olynomials)f Fo(\033)1400 554 y Fk(m)1398 585 y(k)1433 572 y Fp(\()p Fo(a)1478 579 y Fs(1)1497 572 y Fo(;)8 b(a)1545 579 y Fs(2)1564 572 y Fo(;)g(:)g(:)g(:)16 b(;)8 b(a)1708 579 y Fk(k)1729 572 y Fp(\))129 632 y(lab)q(eled)15 b(b)o(y)h(indices)f Fo(m)f Fn(\025)f Fp(0)k(and)g Fo(k)f Fn(\025)d Fp(1.)129 723 y Fh(Prop)r(osition)k(15.)24 b Fg(F)l(or)17 b(al)r(l)i Fo(n)14 b Fn(2)g Fm(Z)814 730 y Fs(+)858 723 y Fg(it)k(holds)f(true)h (that,)g(as)f Fo(x)d Fn(!)f(1)p Fg(,)219 857 y Fo(G)257 836 y Fr(\000)p Fs(1)304 857 y Fp(\()p Fo(x)p Fp(\))h(=)g Fo(x)c Fp(+)h Fo(p)547 864 y Fs(0)568 857 y Fp(\()p Fo(c)p Fp(;)d(ln)o(\()p Fo(x)p Fp(\)\))j(+)840 794 y Fk(n)815 809 y Fl(X)819 915 y Fk(k)q Fs(=1)900 823 y Fo(p)924 830 y Fk(k)946 823 y Fp(\()p Fo(c)p Fp(;)d(ln)o(\()p Fo(x)p Fp(\)\))p 900 845 233 2 v 992 891 a Fo(x)1020 876 y Fk(k)1149 857 y Fp(+)j Fo(O)1236 801 y Fl(\020)1275 786 y(\022)1316 823 y Fp(ln\()p Fo(x)p Fp(\))p 1316 845 107 2 v 1356 891 a Fo(x)1428 786 y Fl(\023)1464 798 y Fk(n)p Fs(+1)1541 801 y Fl(\021)1661 857 y Fp(\(35\))129 999 y Fg(wher)n(e)16 b(the)h(p)n(olynomials)g Fo(p)634 1006 y Fk(n)658 999 y Fp(\()p Fo(c)p Fp(;)8 b Fo(z)r Fp(\))16 b Fg(have)h(b)n(e)n(en)g(de\014ne)n(d)h(in)e(\(7)p (#equation.7) [[369 477 375 489] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)h (\(8)p (#equation.8) [[393 477 399 489] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f(and)h(the)g(c)n(onstant)129 1059 y Fo(c)g Fg(is)g(given)i(by)f(e) n(quality)g(\(32)p (#equation.32) [[218 463 230 475] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)129 1150 y(Pr)n(o)n(of.)k Fp(Corollary)17 b(14)p (#thm.14) [[191 441 203 453] [1 1 1 [3 3]] [0 0 1]] pdfm 17 w(implies)223 1273 y(ln)272 1233 y Fl(\000)295 1273 y Fo(G)333 1253 y Fr(\000)p Fs(1)380 1273 y Fp(\()p Fo(x)p Fp(\))446 1233 y Fl(\001)483 1273 y Fp(=)c(ln\()p Fo(x)p Fp(\))e(+)g Fo(O)748 1203 y Fl(\022)789 1240 y Fp(ln\()p Fo(x)p Fp(\))p 789 1262 V 829 1307 a Fo(x)901 1203 y Fl(\023)946 1273 y Fo(;)1052 1240 y Fp(1)p 989 1262 152 2 v 989 1307 a Fo(G)1027 1293 y Fr(\000)p Fs(1)1074 1307 y Fp(\()p Fo(x)p Fp(\))1159 1273 y(=)1217 1240 y(1)p 1216 1262 28 2 v 1216 1307 a Fo(x)1259 1273 y Fp(+)g Fo(O)1355 1203 y Fl(\022)1397 1240 y Fp(ln\()p Fo(x)p Fp(\))p 1397 1262 107 2 v 1426 1307 a Fo(x)1454 1293 y Fs(2)1508 1203 y Fl(\023)1553 1273 y Fo(:)94 b Fp(\(36\))129 1399 y(Com)o(bining)15 b(\(36)p (#equation.36) [[167 381 179 393] [1 1 1 [3 3]] [0 0 1]] pdfm (\))i (with)f(Lemma)e(12)p (#thm.12) [[256 381 267 393] [1 1 1 [3 3]] [0 0 1]] pdfm 17 w(one)i(deriv)o(es)f(the)h(relation)176 1535 y Fo(x)d Fp(=)h Fo(G)307 1514 y Fr(\000)p Fs(1)355 1535 y Fp(\()p Fo(x)p Fp(\))d Fn(\000)f Fp(3)17 b(ln\()p Fo(G)620 1514 y Fr(\000)p Fs(1)668 1535 y Fp(\()p Fo(x)p Fp(\)\))10 b(+)h Fo(c)g Fn(\000)g Fp(4)952 1473 y Fk(n)927 1488 y Fl(X)931 1594 y Fk(k)q Fs(=1)1012 1501 y Fo(\014)1040 1508 y Fk(k)q Fs(+1)p 1012 1524 94 2 v 1046 1569 a Fo(k)1128 1465 y Fl(\022)1233 1501 y Fp(4)p 1169 1524 152 2 v 1169 1569 a Fo(G)1207 1555 y Fr(\000)p Fs(1)1255 1569 y Fp(\()p Fo(x)p Fp(\))1326 1465 y Fl(\023)1362 1476 y Fk(k)1395 1535 y Fp(+)g Fo(O)1491 1465 y Fl(\022)1568 1501 y Fp(1)p 1532 1524 97 2 v 1532 1569 a Fo(x)1560 1555 y Fk(n)p Fs(+1)1634 1465 y Fl(\023)1679 1535 y Fo(;)1661 1645 y Fp(\(37\))129 1742 y(v)m(alid)18 b(for)h(all)g Fo(n)f Fn(\025)h Fp(0.)29 b(Setting)19 b Fo(n)g Fp(=)f(0)h(in)g(\(37)p (#equation.37) [[304 299 315 311] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h (one)f(arriv)o(es)f(at)h(the)g(case)g Fo(n)f Fp(=)h(0)g(in)129 1802 y(\(35)p (#equation.35) [[108 285 119 297] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)25 b(T)l(o)18 b(\014nish)f(the)g(pro)q(of)i(one)e(can)h(pro)q(ceed,)f(in)g (the)g(ob)o(vious)h(w)o(a)o(y)l(,)e(b)o(y)h(induction)129 1862 y(in)f Fo(n)g Fp(when)g(rep)q(eatedly)g(using)g(relation)g(\(37)p (#equation.37) [[293 270 305 282] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)p 1712 1862 2 33 v 1714 1831 30 2 v 1714 1862 V 1743 1862 2 33 v 129 2005 a Ff(4.4)66 b(Asymptotics)22 b(of)g Fe(h)p Fz(\()p Fe(t)p Fz(\))g Ff(for)g(particular)j(initial)g(data)129 2107 y Fp(W)l(e)17 b(already)h(kno)o(w)f(that)h Fo(h)p Fp(\()p Fo(t)p Fp(\))e(=)h(\()p Fo(G)835 2089 y Fr(\000)p Fs(1)882 2107 y Fp(\(4)p Fo(t)p Fp(\)\))981 2081 y Fs(1)p Fk(=)p Fs(4)1054 2107 y Fp(solv)o(es)g(the)g(problem)f Fo(h)p Fp(\()p Fo(t)p Fp(\))1554 2089 y Fs(3)1574 2107 y Fp(\()p Fo(h)1621 2089 y Fr(0)o(0)1642 2107 y Fp(\()p Fo(t)p Fp(\))c(+)129 2167 y Fo(h)157 2149 y Fr(0)168 2167 y Fp(\()p Fo(t)p Fp(\)\))18 b(=)f(1,)j Fo(h)p Fp(\(0\))e(=)f Fo(h)565 2174 y Fs(0)585 2167 y Fp(,)i Fo(h)646 2149 y Fr(0)658 2167 y Fp(\(0\))f(=)f Fo(h)821 2174 y Fs(1)841 2167 y Fp(.)28 b(Using)19 b(the)f(kno)o(wn)h(asymptotics)e(of)i Fo(G)1634 2149 y Fr(\000)p Fs(1)1682 2167 y Fp(\()p Fo(x)p Fp(\))129 2228 y(w)o(e)c(get)337 2377 y Fo(h)p Fp(\()p Fo(t)p Fp(\))e(=)h(\(4)p Fo(t)p Fp(\))566 2357 y Fs(1)p Fk(=)p Fs(4)629 2292 y Fl( )668 2377 y Fp(1)e(+)778 2315 y Fk(n)753 2330 y Fl(X)757 2436 y Fk(k)q Fs(=1)838 2344 y Fo(p)862 2351 y Fk(k)q Fr(\000)p Fs(1)929 2344 y Fp(\()p Fo(c)p Fp(;)c(ln)o(\(4)p Fo(t)p Fp(\)\))p 838 2366 293 2 v 942 2412 a(4)966 2397 y Fk(k)988 2412 y Fo(t)1006 2397 y Fk(k)1146 2377 y Fp(+)j Fo(O)1242 2307 y Fl(\022)1284 2344 y Fp(ln\()p Fo(t)p Fp(\))1381 2326 y Fk(n)p 1284 2366 120 2 v 1301 2412 a Fo(t)1319 2397 y Fk(n)p Fs(+1)1409 2307 y Fl(\023)1445 2292 y(!)1485 2303 y Fs(1)p Fk(=)p Fs(4)129 2517 y Fp(and)17 b(consequen)o(tly)309 2655 y Fo(h)p Fp(\()p Fo(t)p Fp(\))c(=)h(\(4)p Fo(t)p Fp(\))538 2635 y Fs(1)p Fk(=)p Fs(4)601 2570 y Fl( )641 2655 y Fp(1)d(+)751 2593 y Fk(n)725 2608 y Fl(X)729 2714 y Fk(k)q Fs(=1)811 2622 y Fo(q)833 2629 y Fk(k)q Fr(\000)p Fs(1)899 2622 y Fp(\()p Fo(c)p Fp(;)d(ln)o(\(4)p Fo(t)p Fp(\)\))p 811 2644 290 2 v 936 2689 a Fo(t)954 2675 y Fk(k)1116 2655 y Fp(+)j Fo(O)1203 2600 y Fl(\020)1242 2585 y(\022)1283 2622 y Fp(ln\()p Fo(t)p Fp(\))p 1283 2644 97 2 v 1323 2689 a Fo(t)1385 2585 y Fl(\023)1421 2596 y Fk(n)p Fs(+1)1498 2600 y Fl(\021)1528 2570 y(!)914 2819 y Fp(20)p eop %%Page: 21 21 21 20 bop 129 286 a Fp(where)520 431 y Fo(q)542 438 y Fk(k)576 431 y Fp(=)657 368 y Fk(k)630 383 y Fl(X)628 488 y Fk(m)p Fs(=1)728 397 y Fp(1)p 718 419 46 2 v 718 465 a(4)742 450 y Fk(k)777 360 y Fl(\022)826 377 y Fs(1)p 826 386 18 2 v 826 414 a(4)813 465 y Fo(m)856 360 y Fl(\023)893 431 y Fo(s)916 438 y Fk(m;k)978 431 y Fp(\()p Fo(p)1021 438 y Fs(0)1041 431 y Fo(;)8 b(p)1087 438 y Fs(1)1107 431 y Fo(;)g(:)g(:)g(:)16 b(;)8 b(p)1249 438 y Fk(k)q Fr(\000)p Fs(1)1316 431 y Fp(\))g Fo(:)129 582 y Fp(So)18 b Fo(q)220 589 y Fk(k)260 582 y Fp(are)g(exactly)f(the)h(p)q (olynomials)f(in)o(tro)q(duced)h(in)g(Theorem)f(1)p (#thm.1) [[404 577 410 589] [1 1 1 [3 3]] [0 0 1]] pdfm (.)28 b(It)18 b(is)g(also)h(easy)129 642 y(to)e(see)g(that)h(the)f(degree)f (of)i Fo(q)692 649 y Fk(k)713 642 y Fp(\()p Fo(c)p Fp(;)8 b Fo(z)r Fp(\))17 b(is)g(less)g(than)g(or)h(equal)e(to)i Fo(k)h Fp(since)e(the)g(same)f(is)129 702 y(true)h(for)g(the)g(p)q (olynomials)f Fo(p)689 709 y Fk(n)730 702 y Fp(with)h Fo(n)e Fn(\025)g Fp(1)j(and)g(deg)9 b Fo(p)1184 709 y Fs(0)1219 702 y Fp(=)16 b(1.)24 b(This)17 b(observ)m(ation)h(in)129 762 y(fact)e(completes)e(the)i(pro)q(of)h(of)g(Theorem)e(1)p (#thm.1) [[294 534 299 546] [1 1 1 [3 3]] [0 0 1]] pdfm 17 w(in)g(the)h(case)h(when)f Fo(t)1353 769 y Fs(0)1386 762 y Fp(=)e(0)j(and)f Fo(h)1601 769 y Fs(1)1635 762 y Fo(>)e Fp(0.)129 907 y Ff(4.5)66 b(General)23 b(initial)i(conditions) 129 999 y Fp(Consider)c(\014rst)g(a)h(solution)f Fo(h)p Fp(\()p Fo(t)p Fp(\))g(of)g(\(1)p (#equation.1) [[279 477 284 489] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (with)g(the)g(initial)f(conditions)h Fo(h)p Fp(\(0\))h(=)g Fo(h)1714 1006 y Fs(0)1734 999 y Fp(,)129 1059 y Fo(h)157 1041 y Fr(0)168 1059 y Fp(\(0\))i(=)e Fo(h)342 1066 y Fs(1)362 1059 y Fp(,)g(assuming)g(that)g Fo(h)755 1066 y Fs(1)796 1059 y Fp(is)f(p)q(ositiv)o(e.)36 b(Then,)23 b(as)f(w)o(e)f(already)g(kno)o(w,)i(the)129 1119 y(asymptotic)c(b)q (eha)o(viour)i(of)g Fo(h)p Fp(\()p Fo(t)p Fp(\))g(is)g(describ)q(ed)f (b)o(y)g(Theorem)g(1)p (#thm.1) [[393 448 399 460] [1 1 1 [3 3]] [0 0 1]] pdfm (,)i(i.e.)34 b(equalit)o(y)19 b(\(9)p (#equation.9) [[481 448 487 460] [1 1 1 [3 3]] [0 0 1]] pdfm (\))129 1179 y(holds)d(true)f(with)h Fo(c)e Fp(=)f Fo(c)p Fp(\(0)p Fo(;)8 b(h)669 1186 y Fs(0)689 1179 y Fo(;)g(h)739 1186 y Fs(1)759 1179 y Fp(\).)21 b(Cho)q(ose)c Fo(s)d Fn(2)g Fm(R)f Fp(and)j(set)1288 1166 y(~)1288 1179 y Fo(h)p Fp(\()p Fo(t)p Fp(\))d(=)h Fo(h)p Fp(\()p Fo(t)c Fp(+)g Fo(s)p Fp(\).)21 b(Then)129 1227 y(~)129 1240 y Fo(h)p Fp(\()p Fo(t)p Fp(\))15 b(solv)o(es)h(equation)g(\(1)p (#equation.1) [[212 419 218 431] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g (and)h(satis\014es)f(the)g(initial)f(conditions)1376 1227 y(~)1376 1240 y Fo(h)p Fp(\(0\))f(=)1532 1227 y(~)1532 1240 y Fo(h)1560 1247 y Fs(0)1593 1240 y Fp(=)g Fo(h)p Fp(\()p Fo(s)p Fp(\),)129 1287 y(~)129 1300 y Fo(h)157 1282 y Fr(0)168 1300 y Fp(\(0\))27 b(=)321 1287 y(~)321 1300 y Fo(h)349 1307 y Fs(1)395 1300 y Fp(=)f Fo(h)487 1282 y Fr(0)498 1300 y Fp(\()p Fo(s)p Fp(\).)43 b(But)23 b Fo(h)748 1282 y Fr(0)760 1300 y Fp(\()p Fo(s)p Fp(\))g(is)g(p)q (ositiv)o(e)g(for)g Fo(s)h Fp(su\016cien)o(tly)d(small)h(and)i(so)129 1360 y(equalit)o(y)14 b(\(1)p (#equation.1) [[152 391 158 403] [1 1 1 [3 3]] [0 0 1]] pdfm (\))i (applies)g(to)615 1347 y(~)614 1360 y Fo(h)q Fp(\()p Fo(t)p Fp(\))f(as)i(w)o(ell,)d(with)i Fo(c)g Fp(b)q(eing)g(replaced)f (b)o(y)i(~)-25 b Fo(c)14 b Fp(=)f Fo(c)p Fp(\(0)p Fo(;)1598 1347 y Fp(~)1597 1360 y Fo(h)1625 1367 y Fs(0)1645 1360 y Fo(;)1668 1347 y Fp(~)1667 1360 y Fo(h)1695 1367 y Fs(1)1715 1360 y Fp(\).)129 1420 y(Equating)18 b(the)f(asymptotics)f (of)i Fo(h)p Fp(\()p Fo(t)12 b Fp(+)g Fo(s)p Fp(\))17 b(to)h(the)f(asymptotics)g(of)1415 1407 y(~)1415 1420 y Fo(h)p Fp(\()p Fo(t)p Fp(\))g(one)g(arriv)o(es)129 1480 y(at)f(the)g(equalit)o(y)261 1591 y Fl(\000)284 1631 y Fp(4\()p Fo(t)11 b Fp(+)g Fo(s)p Fp(\))447 1591 y Fl(\001)470 1602 y Fs(1)p Fk(=)p Fs(4)533 1546 y Fl( )572 1631 y Fp(1)h(+)682 1569 y Fk(n)657 1584 y Fl(X)661 1690 y Fk(k)q Fs(=1)742 1594 y Fo(q)764 1601 y Fk(k)785 1594 y Fp(\()p Fo(c)p Fp(;)c(ln)887 1554 y Fl(\000)910 1594 y Fp(4\()p Fo(t)j Fp(+)g Fo(s)p Fp(\))1073 1554 y Fl(\001)1096 1594 y Fp(\))p 742 1620 374 2 v 849 1665 a(\()p Fo(t)f Fp(+)h Fo(s)p Fp(\))987 1651 y Fk(k)1120 1546 y Fl(!)421 1799 y Fp(=)j(\(4)p Fo(t)p Fp(\))553 1778 y Fs(1)p Fk(=)p Fs(4)616 1714 y Fl( )656 1799 y Fp(1)d(+)765 1736 y Fk(n)740 1751 y Fl(X)744 1858 y Fk(k)q Fs(=1)825 1765 y Fo(q)847 1772 y Fk(k)868 1765 y Fp(\()q(~)-25 b Fo(c)p Fp(;)8 b(ln\(4)p Fo(t)p Fp(\)\))p 825 1787 245 2 v 928 1833 a Fo(t)946 1819 y Fk(k)1086 1799 y Fp(+)j Fo(O)1173 1743 y Fl(\020)1211 1729 y(\022)1253 1765 y Fp(ln\()p Fo(t)p Fp(\))p 1253 1787 97 2 v 1292 1833 a Fo(t)1354 1729 y Fl(\023)1391 1740 y Fk(n)p Fs(+1)1468 1743 y Fl(\021)1497 1714 y(!)1661 1716 y Fp(\(38\))129 1954 y(v)m(alid)16 b(for)h Fo(t)d Fn(!)h Fp(+)p Fn(1)i Fp(and)g(ev)o(ery)e Fo(n)g Fn(2)g Fm(Z)876 1961 y Fs(+)903 1954 y Fp(.)23 b(>F)l(rom)15 b(\(38)p (#equation.38) [[338 248 350 260] [1 1 1 [3 3]] [0 0 1]] pdfm (\))j (it)e(is)h(not)g(di\016cult)e(to)i(deriv)o(e)129 2014 y(the)f(relation)g(b)q(et)o(w)o(een)f Fo(c)h Fp(and)i(~)-25 b Fo(c)p Fp(,)16 b(it)g(reads)824 2124 y(~)-25 b Fo(c)14 b Fp(=)g Fo(c)d Fn(\000)g Fp(4)p Fo(s:)608 b Fp(\(39\))129 2234 y(Th)o(us)11 b(the)g(in)o(v)m(ariance)f(of)h(the)g(di\013eren)o (tial)f(equation)g(\(1)p (#equation.1) [[343 181 349 193] [1 1 1 [3 3]] [0 0 1]] pdfm (\))i(is) f(re\015ected)e(in)i(an)h(in)o(v)m(ariance)129 2294 y(of)18 b(the)f(asymptotic)g(expansion)h(of)g(its)f(solutions,)h(as)g (expressed)g(b)o(y)f(relations)g(\(38)p (#equation.38) [[472 166 483 178] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)129 2354 y(\(39)p (#equation.39) [[108 152 119 164] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)22 b(It)16 b(is)g(also)h(clear)e(that)i(these)f(relations)g(m)o(ust)f (hold)h(true)h(not)f(only)g(for)h Fo(s)f Fp(small)129 2415 y(but)g(ev)o(en)f(for)i(all)e Fo(s)f Fn(2)g Fm(R)p Fp(.)202 2475 y(Cho)q(ose)i(no)o(w)e(arbitrary)h(initial)e(data)i(\()p Fo(t)960 2482 y Fs(0)980 2475 y Fo(;)8 b(h)1030 2482 y Fs(0)1049 2475 y Fo(;)g(h)1099 2482 y Fs(1)1119 2475 y Fp(\))14 b Fn(2)g Fm(R)p Fn(\002)7 b Fp(]0)p Fo(;)h Fn(1)p Fp([)g Fn(\002)p Fm(R)g Fp(and)15 b(let)f Fo(h)p Fp(\()p Fo(t)p Fp(\))129 2535 y(b)q(e)g(the)g(corresp)q(onding)i (solution.)21 b(Then,)14 b(as)h(w)o(e)f(kno)o(w)h(from)e(Corollary)i(4) p (#thm.4) [[434 109 440 121] [1 1 1 [3 3]] [0 0 1]] pdfm (,)f Fo(h)1590 2517 y Fr(0)1602 2535 y Fp(\()p Fo(t)p Fp(\))f Fo(>)h Fp(0)129 2595 y(for)21 b(all)f(su\016cien)o(tly)f(large)h Fo(t)p Fp(.)35 b(Fix)20 b Fo(s)h(>)h(t)933 2602 y Fs(0)973 2595 y Fp(suc)o(h)e(that)i Fo(h)1226 2577 y Fr(0)1237 2595 y Fp(\()p Fo(s)p Fp(\))g Fo(>)f Fp(0)g(and)h(set)1605 2582 y(~)1605 2595 y Fo(h)p Fp(\()p Fo(t)p Fp(\))f(=)129 2655 y Fo(h)p Fp(\()p Fo(t)t Fp(+)t Fo(s)p Fp(\).)f(W)l(e)13 b(use)g(once)g(more)f(the)h(already)f(pro)o(v)o(en)h(fact)g(that)1306 2642 y(~)1305 2655 y Fo(h)p Fp(\()p Fo(t)p Fp(\))g(satis\014es)g (equalit)o(y)914 2819 y(21)p eop %%Page: 22 22 22 21 bop 129 286 a Fp(\(9)p (#equation.9) [[108 648 113 660] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)24 b(with)e Fo(c)h Fp(b)q(eing)f(replaced)g(b)o(y)h(~)-25 b Fo(c)24 b Fp(=)h Fo(c)p Fp(\(0)p Fo(;)8 b(h)p Fp(\()p Fo(s)p Fp(\))p Fo(;)g(h)1133 268 y Fr(0)1145 286 y Fp(\()p Fo(s)p Fp(\)\).)39 b(This)23 b(implies)d(that)j(the)129 347 y(asymptotic)15 b(b)q(eha)o(viour)h(of)g Fo(h)p Fp(\()p Fo(t)p Fp(\))e(=)811 333 y(~)811 347 y Fo(h)p Fp(\()p Fo(t)c Fn(\000)h Fo(s)p Fp(\))16 b(is)g(giv)o(en)g(b)o(y)191 485 y Fo(h)p Fp(\()p Fo(t)p Fp(\))e(=)340 445 y Fl(\000)363 485 y Fp(4\()p Fo(t)d Fn(\000)g Fo(s)p Fp(\))527 445 y Fl(\001)550 456 y Fs(1)p Fk(=)p Fs(4)613 400 y Fl( )653 485 y Fp(1)g(+)762 423 y Fk(n)737 438 y Fl(X)741 544 y Fk(k)q Fs(=1)822 448 y Fo(q)844 455 y Fk(k)865 448 y Fp(\()q(~)-25 b Fo(c)p Fp(;)8 b(ln)968 408 y Fl(\000)990 448 y Fp(4\()p Fo(t)k Fn(\000)e Fo(s)p Fp(\))1154 408 y Fl(\001)1177 448 y Fp(\))p 822 474 374 2 v 929 519 a(\()p Fo(t)g Fn(\000)h Fo(s)p Fp(\))1068 505 y Fk(k)1212 485 y Fp(+)g Fo(O)1299 430 y Fl(\020)1338 415 y(\022)1379 451 y Fp(ln\()p Fo(t)p Fp(\))p 1379 474 97 2 v 1419 519 a Fo(t)1481 415 y Fl(\023)1517 426 y Fk(n)p Fs(+1)1594 430 y Fl(\021)1624 400 y(!)1672 485 y Fo(;)129 625 y(n)23 b Fn(2)g Fm(Z)273 632 y Fs(+)299 625 y Fp(.)38 b(But)21 b(in)g(that)h(case)f(one)h(deduces)f(from)g(\(38)p (#equation.38) [[349 567 361 579] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)i (\(39)p (#equation.39) [[379 567 390 579] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f(that)g Fo(h)p Fp(\()p Fo(t)p Fp(\))f(satis\014es)129 685 y(equalit)o(y)14 b(\(9)p (#equation.9) [[152 553 158 565] [1 1 1 [3 3]] [0 0 1]] pdfm (\))j(as) g(w)o(ell,)d(with)i Fo(c)e Fp(=)h(~)-25 b Fo(c)11 b Fp(+)g(4)p Fo(s)p Fp(.)22 b(Theorem)15 b(1)p (#thm.1) [[344 553 350 565] [1 1 1 [3 3]] [0 0 1]] pdfm 16 w(is)h(pro)o(v)o(en.)129 849 y Fq(5)81 b(Additional)38 b(remark:)62 b(comparison)39 b(with)h(the)250 941 y(asymptotics)27 b(of)f Fa(\000)p Fd(W)909 951 y Fn(\000)p Fp(1)975 941 y Fc(\()p Fa(\000)p Fd(e)1089 915 y Fn(\000)p Fo(x)1157 941 y Fc(\))129 1050 y Fp(This)14 b(is)f(a)i(digression)f(whose)h(aim)d (is)i(to)g(emphasize)e(a)i(rather)g(close)g(similarit)o(y)d(of)j(the) 129 1110 y(asymptotic)f(b)q(eha)o(viour)i(of)g(the)f(function)h Fo(G)967 1092 y Fr(\000)p Fs(1)1014 1110 y Fp(\()p Fo(x)p Fp(\))g(with)f(that)h(of)g(Lam)o(b)q(ert)f(function.)129 1170 y(The)19 b(Lam)o(b)q(ert)g(function)g Fo(W)7 b Fp(\()p Fo(z)r Fp(\))19 b(giv)o(es)f(the)h(principal)g(solution)g(for)h Fo(w)h Fp(in)e Fo(z)h Fp(=)g Fo(w)q(e)1720 1152 y Fk(w)129 1231 y Fp(and)c Fo(W)269 1238 y Fk(k)290 1231 y Fp(\()p Fo(z)r Fp(\))f(giv)o(es)f(the)h Fo(k)596 1213 y Fs(th)647 1231 y Fp(solution.)21 b(Surprisingly)l(,)15 b(it)f(is)h(not)h(do)q (cumen)o(ted)e(in)h(some)129 1291 y(standard)g(text)e(b)q(o)q(oks)i (and)f(reference)e(b)q(o)q(oks)j(on)f(sp)q(ecial)g(functions)f(though)i (w)o(e)e(ma)o(y)129 1351 y(ha)o(v)o(e)j(missed)g(some)g(sources.)25 b(On)18 b(the)f(other)g(hand,)h(the)f(Lam)o(b)q(ert)g(function)g(seems) 129 1411 y(to)e(ha)o(v)o(e)g(attracted)g(ev)o(en)g(in)g(a)g(rather)h (recen)o(t)e(p)q(erio)q(d)i(some)e(atten)o(tion,)h(particularly)129 1471 y(from)h(the)h(computational)g(and)h(com)o(binatorial)e(p)q(oin)o (t)i(of)g(view)e(\(see)i([3)p (#cite.LambertW) [[425 364 431 376] [1 1 1 [3 3]] [0 0 1]] pdfm (])f (for)g(a)h(sum-)129 1532 y(mary\).)28 b(It)18 b(is)h(also)g(implem)o (en)o(te)o(d)d(in)j(some)f(standard)i(computer)d(algebra)j(systems)129 1592 y(lik)o(e)15 b(Maple)h(and)h(Mathematica)e(where)h(it)h(is)f (called)g(Lam)o(b)q(ertW)g(and)h(Pro)q(ductLog,)129 1652 y(resp)q(ectiv)o(ely)l(.)k(Let)d(us)f(just)h(brie\015y)e(recall)g(that) i Fo(W)7 b Fp(\()p Fo(z)r Fp(\))17 b(is)g(analytic)f(in)h(a)h(neigh)o (b)q(our-)129 1712 y(ho)q(o)q(d)g(of)e Fo(z)g Fp(=)e(0)i(with)g(the)g (con)o(v)o(ergence)f(radius)i(equal)e(to)i Fo(e)1260 1694 y Fr(\000)p Fs(1)1307 1712 y Fp(,)656 1842 y Fo(W)7 b Fp(\()p Fo(z)r Fp(\))14 b(=)856 1780 y Fr(1)838 1795 y Fl(X)841 1901 y Fk(k)q Fs(=1)923 1808 y Fp(\()p Fn(\000)p Fp(1\))1024 1790 y Fk(k)1045 1808 y Fo(k)1072 1790 y Fk(k)q Fr(\000)p Fs(1)p 923 1831 216 2 v 1010 1876 a Fo(k)r Fp(!)1152 1842 y Fo(z)1177 1822 y Fk(k)1206 1842 y Fo(:)129 1979 y Fp(The)k(co)q(e\016cien)o(ts)e(ha)o(v)o(e)i(a)g(com)o (binatorial)e(in)o(terpretation)h(when)i(coun)o(ting)e(distinct)129 2039 y(orien)o(ted)e(trees.)202 2100 y(Consider)h(no)o(w)h(the)f (equation)789 2194 y Fo(y)d Fn(\000)e Fp(ln\()p Fo(y)r Fp(\))i(=)h Fo(x;)129 2289 y Fp(or,)i(equiv)m(alen)o(tly)l(,)810 2384 y Fo(y)r(e)859 2364 y Fr(\000)p Fk(y)920 2384 y Fp(=)e Fo(e)995 2364 y Fr(\000)p Fk(x)1052 2384 y Fo(:)129 2479 y Fp(It)j(is)h(elemen)o(tary)d(to)k(see)f(that)g(for)g Fo(x)f Fn(2)8 b Fp(]1)p Fo(;)g Fp(+)p Fn(1)p Fp([)18 b(there)f(are)i(exactly)d(t)o(w)o(o)i(real)g(solu-)129 2539 y(tions,)j Fo(y)291 2546 y Fs(1)311 2539 y Fp(\()p Fo(x)p Fp(\))f(and)h Fo(y)520 2546 y Fs(2)540 2539 y Fp(\()p Fo(x)p Fp(\),)g(with)f Fo(y)780 2546 y Fs(1)800 2539 y Fp(\()p Fo(x)p Fp(\))h Fn(2)8 b Fp(]0)p Fo(;)g Fp(1[)29 b(and)21 b Fo(y)1178 2546 y Fs(2)1198 2539 y Fp(\()p Fo(x)p Fp(\))g Fn(2)8 b Fp(]1)p Fo(;)g Fp(+)p Fn(1)p Fp([)g(.)34 b(The)21 b(b)q(oth)129 2599 y(solutions)16 b(can)h(b)q(e)f(expressed)g(with)g(the)g(aid)g(of)h(the)f(Lam)o(b)q (ert)f(function,)h(namely)457 2694 y Fo(y)481 2701 y Fs(1)500 2694 y Fp(\()p Fo(x)p Fp(\))e(=)f Fn(\000)p Fo(W)7 b Fp(\()p Fn(\000)p Fo(e)804 2673 y Fr(\000)p Fk(x)853 2694 y Fp(\))p Fo(;)56 b(y)966 2701 y Fs(2)986 2694 y Fp(\()p Fo(x)p Fp(\))13 b(=)h Fn(\000)p Fo(W)1202 2701 y Fr(\000)p Fs(1)1249 2694 y Fp(\()p Fn(\000)p Fo(e)1330 2673 y Fr(\000)p Fk(x)1379 2694 y Fp(\))8 b Fo(:)914 2819 y Fp(22)p eop %%Page: 23 23 23 22 bop 202 286 a Fp(The)14 b(aim)f(of)h(this)g(remark)f(is)g(to)i(p) q(oin)o(t)f(out)h(that)f(the)g(asymptotics)f(of)i(the)e(second)129 347 y(solution,)f(i.e.)19 b Fn(\000)p Fo(W)491 354 y Fr(\000)p Fs(1)538 347 y Fp(\()p Fn(\000)p Fo(e)619 329 y Fr(\000)p Fk(x)668 347 y Fp(\),)12 b(as)g Fo(x)i Fn(!)g Fp(+)p Fn(1)p Fp(,)e(can)g(b)q(e)g(deriv)o(ed)e(in)i(a)g(w)o(a)o(y)g (quite)f(similar)129 407 y(to)17 b(what)g(w)o(e)f(ha)o(v)o(e)g(done)h (in)g(Subsection)f(4.3)p (#subsection.4.3) [[292 619 306 631] [1 1 1 [3 3]] [0 0 1]] pdfm 17 w(when)h(treating)g(the)f(function)h Fo(G)1621 389 y Fr(\000)p Fs(1)1668 407 y Fp(\()p Fo(x)p Fp(\).)129 467 y(T)l(o)f(this)h(end)f(let)f(us)i(recursiv)o(ely)d(de\014ne)h(p)q (olynomials)20 b(~)-28 b Fo(p)1200 474 y Fk(k)1222 467 y Fp(\()p Fo(z)r Fp(\),)415 549 y(~)g Fo(p)435 556 y Fs(0)455 549 y Fp(\()p Fo(z)r Fp(\))14 b(=)g Fo(z)r(;)28 b Fp(~)-28 b Fo(p)671 556 y Fk(k)q Fs(+1)738 549 y Fp(\()p Fo(z)r Fp(\))13 b(=)h Fo(\033)896 528 y Fs(0)894 561 y Fk(k)q Fs(+1)960 549 y Fp(\()t(~)-28 b Fo(p)1003 556 y Fs(0)1023 549 y Fp(\()p Fo(z)r Fp(\))p Fo(;)12 b Fp(~)-28 b Fo(p)1132 556 y Fs(1)1152 549 y Fp(\()p Fo(z)r Fp(\))p Fo(;)8 b(:)g(:)g(:)f(;)12 b Fp(~)-28 b Fo(p)1348 556 y Fk(k)1370 549 y Fp(\()p Fo(z)r Fp(\)\))p Fo(:)129 630 y Fp(F)l(or)20 b Fo(k)j Fn(\025)e Fp(1,)h(the)e(degree)g(of)h(the)f(p)q (olynomial)k(~)-29 b Fo(p)1060 637 y Fk(k)1082 630 y Fp(\()p Fo(z)r Fp(\))21 b(is)f Fo(k)r Fp(.)34 b(Here)19 b(are)i(sev)o(eral)e(\014rst)129 690 y(p)q(olynomials:)319 791 y(~)-28 b Fo(p)339 798 y Fs(1)359 791 y Fp(\()p Fo(z)r Fp(\))14 b(=)g Fo(z)r(;)28 b Fp(~)-28 b Fo(p)575 798 y Fs(2)595 791 y Fp(\()p Fo(z)r Fp(\))14 b(=)f Fo(z)g Fn(\000)814 758 y Fp(1)p 814 780 25 2 v 814 826 a(2)852 791 y Fo(z)877 771 y Fs(2)896 791 y Fo(;)29 b Fp(~)-29 b Fo(p)958 798 y Fs(3)979 791 y Fp(\()p Fo(z)r Fp(\))13 b(=)h Fo(z)f Fn(\000)1198 758 y Fp(3)p 1198 780 V 1198 826 a(2)1235 791 y Fo(z)1260 771 y Fs(2)1291 791 y Fp(+)1345 758 y(1)p 1345 780 V 1345 826 a(3)1383 791 y Fo(z)1408 771 y Fs(3)1427 791 y Fo(;)33 b(:)8 b(:)g(:)16 b(:)129 903 y Fh(Prop)r(osition)h(16.)24 b Fg(It)18 b(holds,)f(as)h Fo(x)13 b Fn(!)h Fp(+)p Fn(1)p Fg(,)645 985 y Fn(\000)p Fo(W)730 992 y Fr(\000)p Fs(1)776 985 y Fp(\()p Fn(\000)p Fo(e)857 964 y Fr(\000)p Fk(x)906 985 y Fp(\))g(=)g Fo(x)d Fp(+)g Fo(O)q Fp(\(ln)d Fo(x)p Fp(\))129 1066 y Fg(and,)17 b(for)g Fo(n)d Fn(\025)g Fp(0)p Fg(,)378 1189 y Fn(\000)p Fo(W)463 1196 y Fr(\000)p Fs(1)510 1189 y Fp(\()p Fn(\000)p Fo(e)591 1168 y Fr(\000)p Fk(x)639 1189 y Fp(\))g(=)g Fo(x)d Fp(+)837 1126 y Fk(n)812 1141 y Fl(X)816 1247 y Fk(k)q Fs(=0)901 1155 y Fp(~)-28 b Fo(p)921 1162 y Fk(k)943 1155 y Fp(\(ln)8 b Fo(x)p Fp(\))p 897 1177 161 2 v 953 1223 a Fo(x)981 1208 y Fk(k)1073 1189 y Fp(+)j Fo(O)1169 1103 y Fl( )1209 1118 y(\022)1250 1155 y Fp(ln)d Fo(x)p 1250 1177 77 2 v 1275 1223 a(x)1332 1118 y Fl(\023)1369 1130 y Fk(n)p Fs(+1)1437 1103 y Fl(!)1485 1189 y Fo(:)202 1312 y Fp(The)17 b(prop)q(osition)h(can)f(b)q(e)g(pro)o(v)o(en)f(using) h(a)h(similar)d(approac)o(h)i(as)h(the)e(one)i(used)129 1373 y(in)e(the)h(pro)q(of)g(of)g(Lemma)e(13)p (#thm.13) [[223 387 234 399] [1 1 1 [3 3]] [0 0 1]] pdfm (.)24 b(In)16 b(fact,)g(this)h(asymptotic)e(expansion)i(is)g(w)o(ell)e(kno)o (wn)129 1433 y(and)k(is)g(in)g(agreemen)o(t)e(with)i(what)g(has)h(b)q (een)f(published)f(in)h([4)p (#cite.Bruijn) [[388 373 394 385] [1 1 1 [3 3]] [0 0 1]] pdfm (,)g(5)p (#cite.Comtet) [[402 373 408 385] [1 1 1 [3 3]] [0 0 1]] pdfm (])g (and)g([3)p (#cite.LambertW) [[442 373 448 385] [1 1 1 [3 3]] [0 0 1]] pdfm (])g (though)129 1493 y(the)d(deriv)m(ation)g(and)g(presen)o(tation)h(here)e (is)h(somewhat)g(di\013eren)o(t.)129 1653 y Fh(Ac)n(kno)n(wledgemen)n (ts.)33 b Fp(R.D.B.)19 b(wishes)i(to)f(thank)h(F)o(ONDECYT)f(\(Chile\)) g(199{)129 1713 y(0427,)d(and)g(the)f(action)g(C94E10)i(of)f (ECOS-CONICYT.)f(P)l(.)1277 1700 y(\024)1276 1713 y(S.)f(wishes)i(to)f (gratefully)129 1773 y(ac)o(kno)o(wledge)22 b(the)g(partial)h(supp)q (ort)h(from)e(Gran)o(t)h(No.)41 b(201/01/013)q(08)26 b(of)d(Gran)o(t)129 1833 y(Agency)15 b(of)i(the)f(Czec)o(h)f(Republic.) 129 1996 y Fq(References)129 2105 y Fp([1])24 b(Abraham)13 b(R.,)h(Marsden)h(J.E.:)20 b Fg(F)l(oundations)d(of)f(Me)n(chanics)p Fp(.)j(Addison-W)l(esley)l(,)205 2165 y(1978.)129 2259 y([2])24 b(Halp)q(erin)15 b(B.:)26 b(Ph)o(ys.)15 b(Rev.)h(B)f Fh(25)i Fp(\(1982\))g(2185\013.)129 2353 y([3])24 b(Corless)17 b(R.M.,)e(Gonnet)i(G.H.,)f(Hare)g(D.E.G.,)g(Je\013rey)g(D.J.,)g(Kn)o (uth)h(D.E.:)22 b Fg(On)205 2413 y(the)h(L)n(amb)n(ert)e(W)h(F)l (unction)p Fp(.)40 b(Adv)m(ances)21 b(in)h(Computational)f(Mathematics) g Fh(5)205 2473 y Fp(\(1996\))c(329-359.)129 2567 y([4])24 b(de)16 b(Bruijn)f(N.G.:)20 b Fg(Asymptotic)e(Metho)n(ds)f(in)g(A)o (nalysis)p Fp(.)22 b(North-Holland,)15 b(1961.)129 2660 y([5])24 b(Com)o(tet)14 b(L.:)27 b(C.)16 b(R.)g(Acad.)f(Sc.)h(P)o(aris) g Fh(270)g Fp(\(1970\))i(1085-1088.)914 2819 y(23)p eop %%Trailer end end userdict /end-hook known{end-hook}if %%EOF ---------------0110240829350--