Content-Type: multipart/mixed; boundary="-------------0305171006305" This is a multi-part message in MIME format. ---------------0305171006305 Content-Type: text/plain; name="03-224.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-224.comments" PostScript file, 32 pages, 1 figure ---------------0305171006305 Content-Type: text/plain; name="03-224.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-224.keywords" Quantum Knizhnik-Zamolodchikov equations, Heisenberg XXX spin chain, correlation functions, cohomologies of deformed Jacobi varieties, emptiness formation probability. ---------------0305171006305 Content-Type: application/postscript; name="s.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="s.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: second1.dvi %%Pages: 32 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: Times-Bold Times-Roman Times-Italic %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o s.ps second1.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.05.17:1043 %%BeginProcSet: texc.pro %! 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Fl(l)2185 3969 y Fj(\033)2240 3984 y Fl(k)r Fk(\000)p Fl(l)2360 3969 y Fi(\()p Fj(\025)2455 3984 y Fm(1)2494 3969 y Fj(;)17 b(:)g(:)g(:)f(;)h(\025)2770 3984 y Fl(n)2816 3969 y Fi(\))234 4271 y Fy(Actually)-6 b(,)27 b(the)g(multipliers)1233 4196 y Fn(Q)1327 4271 y Fi(\()p Fj(\013)e Fo(\000)f Fj(\014)1608 4286 y Fl(j)1669 4271 y Fo(\006)1781 4232 y Fl(\031)r(i)p 1781 4248 67 4 v 1797 4305 a Fm(2)1858 4271 y Fi(\))j Fy(in)h(\(36\))f(cancel)h(singularities)e(of)i Fj(')p Fi(\()p Fj(\013)d Fo(\000)f Fj(\014)3475 4286 y Fl(j)3512 4271 y Fi(\))j Fy(closest)g(to)h(the)88 4391 y(real)d(axis.)30 b(So,)25 b(the)g(pinching)e(does)i(not)f(tak)o(e)h(place)g(and,)g (hence,)g(the)g(result)f(is)g(\002nite.)31 b(Denote:)1398 4598 y Fj(I)1441 4613 y Fl(k)r(;m)1593 4598 y Fi(=)d Fj(I)1748 4557 y Fm(0)1740 4623 y Fl(k)r(;m)1886 4598 y Fi(+)1984 4504 y Fn(X)2145 4598 y Fj(i\016)2221 4613 y Fl(p)2261 4598 y Fj(I)2312 4551 y Fl(p)2304 4626 y(k)r(;m)2451 4598 y Fi(+)22 b Fo(O)s Fi(\()p Fj(\016)2716 4557 y 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Fm(=1)1607 5390 y Fi(4)p Fj(\031)1715 5354 y Fm(3)p 1417 5434 528 4 v 1417 5525 a Fi(sinh)o(\()p Fj(\013)23 b Fo(\000)g Fj(\025)1870 5540 y Fl(j)1906 5525 y Fi(\))1954 5457 y Fj(e)1999 5416 y Fm(\()p Fl(n)p Fk(\000)p Fm(1)p Fk(\000)p Fm(2)p Fl(m)p Fm(\))p Fl(\013)2389 5457 y Fj(Q)2466 5472 y Fl(k)2509 5457 y Fi(\()p Fj(\013)g Fi(+)2740 5390 y Fj(\031)t(i)p 2740 5434 93 4 v 2762 5525 a Fi(2)2842 5457 y Fo(j)p Fj(\025)2927 5472 y Fm(1)2966 5457 y Fj(;)17 b Fo(\001)g(\001)g(\001)31 b Fj(;)17 b(\025)3260 5472 y Fl(n)3307 5457 y Fi(\))28 b(=)f(0)2046 5871 y Fy(11)p eop %%Page: 12 12 12 11 bop 88 289 a Fy(So,)25 b(the)f(leading)g(contrib)n(ution)f(is)i (of)g(the)f(\002rst)h(order)g(in)g Fj(\016)t Fy(')-5 b(s.)30 b(Consider)25 b Fj(I)2686 241 y Fl(p)2678 316 y(k)r(;m)2802 289 y Fy(:)221 522 y Fj(I)272 474 y Fl(p)264 549 y(k)r(;m)416 522 y Fi(=)268 623 y Fn(Z)215 907 y Fb(R)p Fm(+)p Fl(i)p Fm(0)416 759 y Fj(d\013)588 634 y Fl(n)546 664 y Fn(Y)548 874 y Fl(j)t Fm(=1)890 691 y Fi(4)p Fj(\031)998 655 y Fm(3)p 700 736 528 4 v 700 827 a Fi(sinh\()p Fj(\013)d Fo(\000)h Fj(\025)1153 842 y Fl(j)1190 827 y Fi(\))1237 759 y Fj(e)1282 717 y Fm(\()p Fl(n)p Fk(\000)p Fm(1)p Fk(\000)p Fm(2)p Fl(m)p Fm(\))p Fl(\013)1689 618 y Fn(\022)1763 759 y Fj(\037)p Fi(\()p Fj(\013)f Fo(\000)h Fj(\025)2103 774 y Fl(p)2143 759 y Fi(\))f Fo(\000)2428 691 y Fi(1)p 2312 736 282 4 v 2312 827 a Fj(\013)h Fo(\000)g Fj(\025)2554 842 y Fl(p)2625 759 y Fo(\000)2783 691 y Fj(@)p 2735 736 154 4 v 2735 827 a(@)5 b(\025)2848 842 y Fl(p)2898 618 y Fn(\023)2988 759 y Fj(Q)3065 774 y Fl(k)3108 759 y Fi(\()p Fj(\013)23 b Fo(\000)3341 691 y Fj(\031)t(i)p 3341 736 93 4 v 3362 827 a Fi(2)3443 759 y Fo(j)p Fj(\025)3528 774 y Fm(1)3567 759 y Fj(;)17 b Fo(\001)g(\001)g(\001)31 b Fj(;)17 b(\025)3861 774 y Fl(n)3908 759 y Fi(\))24 b(+)145 1120 y(+)268 984 y Fn(Z)215 1268 y Fb(R)p Fk(\000)p Fl(i)p Fm(0)416 1120 y Fj(d\013)588 995 y Fl(n)546 1025 y Fn(Y)548 1235 y Fl(j)t Fm(=1)890 1052 y Fi(4)p Fj(\031)998 1016 y Fm(3)p 700 1097 528 4 v 700 1188 a Fi(sinh\()p Fj(\013)e Fo(\000)h Fj(\025)1153 1203 y Fl(j)1190 1188 y Fi(\))1237 1120 y Fj(e)1282 1079 y Fm(\()p Fl(n)p Fk(\000)p Fm(1)p Fk(\000)p Fm(2)p Fl(m)p Fm(\))p Fl(\013)1689 979 y Fn(\022)1763 1120 y Fj(\037)p Fi(\()p Fj(\013)f Fo(\000)h Fj(\025)2103 1135 y Fl(p)2143 1120 y Fi(\))f(+)2427 1052 y(1)p 2311 1097 282 4 v 2311 1188 a Fj(\013)g Fo(\000)h Fj(\025)2552 1203 y Fl(p)2624 1120 y Fi(+)2780 1052 y Fj(@)p 2732 1097 154 4 v 2732 1188 a(@)5 b(\025)2845 1203 y Fl(p)2895 979 y Fn(\023)2985 1120 y Fj(Q)3062 1135 y Fl(k)3105 1120 y Fi(\()p Fj(\013)23 b Fi(+)3336 1052 y Fj(\031)t(i)p 3336 1097 93 4 v 3357 1188 a Fi(2)3438 1120 y Fo(j)p Fj(\025)3523 1135 y Fm(1)3562 1120 y Fj(;)17 b Fo(\001)g(\001)g(\001)31 b Fj(;)17 b(\025)3856 1135 y Fl(n)3903 1120 y Fi(\))88 1461 y Fy(where)921 1733 y Fj(\037)p Fi(\()p Fj(\013)q Fi(\))27 b(=)1293 1666 y Fj(d)p 1262 1710 114 4 v 1262 1802 a(d\013)1402 1593 y Fn(\022)1475 1733 y Fi(log)1628 1660 y Fj(')p Fi(\()p Fj(\013)22 b 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2389 y Fm(1)2362 2374 y Fj(;)17 b Fo(\001)g(\001)g(\001)31 b Fj(;)17 b(\025)2656 2389 y Fl(n)2703 2374 y Fi(\))p Fj(J)2804 2326 y Fl(p)2795 2401 y(l)q(;m)88 2677 y Fy(Then)163 2897 y Fj(J)226 2850 y Fl(p)217 2925 y(l)q(;m)353 2897 y Fi(=)211 3010 y Fn(Z)158 3293 y Fb(R)p Fm(+)p Fl(i)p Fm(0)359 3145 y Fj(d\013)531 3021 y Fl(n)489 3051 y Fn(Y)491 3261 y Fl(j)t Fm(=1)833 3078 y Fi(4)p Fj(\031)941 3042 y Fm(3)p 643 3122 528 4 v 643 3214 a Fi(sinh\()p Fj(\013)22 b Fo(\000)h Fj(\025)1096 3229 y Fl(j)1132 3214 y Fi(\))1180 3145 y Fj(e)1225 3104 y Fm(\()p Fl(n)p Fk(\000)p Fm(1)p Fk(\000)p Fm(2)p Fl(m)p Fm(\))p Fl(\013)1632 2975 y Fn( )1711 3145 y Fj(\037)p Fi(\()p Fj(\013)g Fo(\000)f Fj(\025)2051 3160 y Fl(p)2091 3145 y Fi(\))2146 3065 y Fn(\000)2191 3145 y Fi(\()p Fj(\013)h Fo(\000)g Fj(\031)t(i)p Fi(\))2544 3104 y Fl(l)2592 3145 y Fo(\000)g Fj(\013)2755 3104 y Fl(l)2780 3065 y Fn(\001)2848 3145 y Fi(+)f Fj(\031)t(i)3214 3069 y Fi(\()p Fj(\013)g Fo(\000)h Fj(\031)t(i)p Fi(\))3566 3033 y Fl(l)3614 3069 y Fo(\000)g Fj(\025)3771 3033 y Fl(l)3771 3093 y(p)p 3048 3122 928 4 v 3048 3214 a Fi(\()p Fj(\013)g Fo(\000)g Fj(\025)3328 3229 y Fl(p)3367 3214 y Fi(\)\()p Fj(\013)g Fo(\000)f Fj(\025)3684 3229 y Fl(p)3746 3214 y Fo(\000)h Fj(\031)t(i)p Fi(\))3986 2975 y Fn(!)4106 3145 y Fi(+)88 3518 y(+)211 3382 y Fn(Z)158 3666 y Fb(R)p Fk(\000)p Fl(i)p Fm(0)359 3518 y Fj(d\013)531 3393 y Fl(n)489 3423 y Fn(Y)491 3633 y Fl(j)t Fm(=1)833 3450 y Fi(4)p Fj(\031)941 3414 y Fm(3)p 643 3495 528 4 v 643 3586 a Fi(sinh\()p Fj(\013)f Fo(\000)h Fj(\025)1096 3601 y Fl(j)1132 3586 y Fi(\))1180 3518 y Fj(e)1225 3477 y Fm(\()p Fl(n)p Fk(\000)p Fm(1)p Fk(\000)p Fm(2)p Fl(m)p Fm(\))p Fl(\013)1632 3347 y Fn( )1711 3518 y Fj(\037)p Fi(\()p Fj(\013)g Fo(\000)f Fj(\025)2051 3533 y Fl(p)2091 3518 y Fi(\))2146 3437 y Fn(\000)2191 3518 y Fj(\013)2254 3477 y Fl(l)2302 3518 y Fo(\000)h Fi(\()p Fj(\013)f Fi(+)g Fj(\031)t(i)p Fi(\))2752 3477 y Fl(l)2779 3437 y Fn(\001)2846 3518 y Fo(\000)h Fj(\031)t(i)3214 3441 y Fi(\()p Fj(\013)f Fi(+)g Fj(\031)t(i)p Fi(\))3564 3405 y Fl(l)3613 3441 y Fo(\000)g Fj(\025)3769 3405 y Fl(l)3769 3466 y(p)p 3048 3495 927 4 v 3048 3586 a Fi(\()p Fj(\013)h Fo(\000)g Fj(\025)3328 3601 y Fl(p)3367 3586 y Fi(\)\()p Fj(\013)g Fo(\000)f Fj(\025)3684 3601 y Fl(p)3746 3586 y Fi(+)g Fj(\031)t(i)p Fi(\))3984 3347 y Fn(!)88 3859 y Fy(Here)j(we)g(ha)n(v)o (e)g(used)f(the)h(follo)n(wing)e(identity)1106 4101 y Fj(@)p 1058 4146 154 4 v 1058 4237 a(@)5 b(\025)1171 4252 y Fl(p)1221 4169 y Fj(\033)1276 4184 y Fl(k)1319 4169 y Fi(\()p Fj(\025)1414 4184 y Fm(1)1454 4169 y Fj(;)17 b Fo(\001)g(\001)g(\001)31 b Fj(;)17 b(\025)1748 4184 y Fl(n)1794 4169 y Fi(\))28 b(=)1971 4044 y Fl(k)r Fk(\000)p Fm(1)1964 4074 y Fn(X)1979 4286 y Fl(l)q Fm(=0)2107 4169 y Fi(\()p Fo(\000)p Fj(\025)2279 4184 y Fl(p)2319 4169 y Fi(\))2357 4128 y Fl(k)r Fk(\000)p Fl(l)q Fk(\000)p Fm(1)2567 4169 y Fj(\033)2622 4184 y Fl(l)2648 4169 y Fi(\()p Fj(\025)2743 4184 y Fm(1)2783 4169 y Fj(;)17 b(:)g(:)g(:)e(;)i(\025)3058 4184 y Fl(n)3105 4169 y Fi(\))234 4472 y Fy(Let)25 b(us)f(denote)1778 4593 y Fj(f)11 b Fi(\()p Fj(\013)q Fi(\))27 b(=)g Fj(\013)2169 4551 y Fl(l)2217 4593 y Fo(\000)c Fj(\025)2374 4551 y Fl(l)2374 4617 y(p)88 4767 y Fy(then)h(using)g(the)g(identities:)1413 5028 y Fj(\037)p Fi(\()p Fj(\013)f Fi(+)f Fj(\031)t(i)p Fi(\))g(+)g Fj(\037)p Fi(\()p Fj(\013)q Fi(\))27 b(=)2399 4961 y(1)p 2286 5005 276 4 v 2286 5096 a Fj(\013)22 b Fi(+)g Fj(\031)t(i)2593 5028 y Fo(\000)2709 4961 y Fi(1)p 2702 5005 63 4 v 2702 5096 a Fj(\013)3937 5028 y Fy(\(38\))1413 5269 y Fj(\037)p Fi(\()p Fj(\013)q Fi(\))f(+)h Fj(\037)p Fi(\()p Fj(\013)h Fo(\000)g Fj(\031)t(i)p Fi(\))28 b(=)2294 5201 y(1)p 2287 5246 V 2287 5337 a Fj(\013)2382 5269 y Fo(\000)2606 5201 y Fi(1)p 2492 5246 277 4 v 2492 5337 a Fj(\013)22 b Fo(\000)h Fj(\031)t(i)2046 5871 y Fy(12)p eop %%Page: 13 13 13 12 bop 88 289 a Fy(one)24 b(\002nds:)251 559 y Fj(J)314 512 y Fl(p)305 587 y(l)q(;m)441 559 y Fi(=)564 423 y Fn(Z)511 707 y Fb(R)p Fm(+)p Fl(i)p Fm(0)712 559 y Fj(d\013)884 434 y Fl(n)842 464 y Fn(Y)844 674 y Fl(j)t Fm(=1)1186 492 y Fi(4)p Fj(\031)1294 455 y Fm(3)p 996 536 528 4 v 996 627 a Fi(sinh\()p Fj(\013)e Fo(\000)h Fj(\025)1449 642 y Fl(j)1485 627 y Fi(\))1533 559 y Fj(e)1578 518 y Fm(\()p Fl(n)p Fk(\000)p Fm(1)p Fk(\000)p Fm(2)p Fl(m)p Fm(\))p Fl(\013)1985 559 y Fi(\()p Fj(f)11 b Fi(\()p Fj(\013)23 b Fo(\000)f Fj(\031)t(i)p Fi(\))p Fj(\037)p Fi(\()p Fj(\013)h Fo(\000)g Fj(\025)2775 574 y Fl(p)2836 559 y Fo(\000)g Fj(\031)t(i)p Fi(\))f(+)g Fj(f)11 b Fi(\()p Fj(\013)q Fi(\))p Fj(\037)p Fi(\()p Fj(\013)23 b Fo(\000)f Fj(\025)3724 574 y Fl(p)3764 559 y Fi(\)\))16 b Fo(\000)439 920 y(\000)564 785 y Fn(Z)511 1068 y Fb(R)p Fk(\000)p Fl(i)p Fm(0)712 920 y Fj(d\013)884 796 y Fl(n)842 826 y Fn(Y)844 1036 y Fl(j)t Fm(=1)1186 853 y Fi(4)p Fj(\031)1294 817 y Fm(3)p 996 897 V 996 989 a Fi(sinh\()p Fj(\013)22 b Fo(\000)h Fj(\025)1449 1004 y Fl(j)1485 989 y Fi(\))1533 920 y Fj(e)1578 879 y Fm(\()p Fl(n)p Fk(\000)p Fm(1)p Fk(\000)p Fm(2)p Fl(m)p Fm(\))p Fl(\013)1985 920 y Fi(\()p Fj(f)11 b Fi(\()p Fj(\013)23 b Fi(+)f Fj(\031)t(i)p Fi(\))p Fj(\037)p Fi(\()p Fj(\013)h Fo(\000)f Fj(\025)2773 935 y Fl(p)2835 920 y Fi(+)g Fj(\031)t(i)p Fi(\))g(+)g Fj(f)11 b Fi(\()p Fj(\013)q Fi(\))p Fj(\037)p Fi(\()p Fj(\013)22 b Fo(\000)h Fj(\025)3721 935 y Fl(p)3760 920 y Fi(\)\))28 b(=)544 1323 y(=)g(2)714 1122 y Fn(0)714 1302 y(@)831 1187 y(Z)778 1471 y Fb(R)p Fm(+)p Fl(i)p Fm(0)979 1323 y Fo(\000)1104 1187 y Fn(Z)1050 1471 y Fb(R)p Fk(\000)p Fl(i)p Fm(0)1235 1122 y Fn(1)1235 1302 y(A)1339 1323 y Fj(d\013)1511 1198 y Fl(n)1469 1228 y Fn(Y)1471 1438 y Fl(j)t Fm(=1)1813 1255 y Fi(4)p Fj(\031)1921 1219 y Fm(3)p 1623 1300 V 1623 1391 a Fi(sinh\()p Fj(\013)23 b Fo(\000)f Fj(\025)2076 1406 y Fl(j)2113 1391 y Fi(\))2161 1323 y Fj(e)2206 1282 y Fm(\()p Fl(n)p Fk(\000)p Fm(1)p Fk(\000)p Fm(2)p Fl(m)p Fm(\))p Fl(\013)2596 1323 y Fj(\037)p Fi(\()p Fj(\013)h Fo(\000)f Fj(\025)2936 1338 y Fl(p)2976 1323 y Fi(\))3031 1242 y Fn(\000)3076 1323 y Fj(\013)3139 1282 y Fl(l)3187 1323 y Fo(\000)g Fj(\025)3343 1282 y Fl(l)3343 1347 y(p)3383 1242 y Fn(\001)3456 1323 y Fi(=)544 1667 y(=)28 b(4)p Fj(\031)t(i)p Fi(\(4)p Fj(\031)935 1626 y Fm(3)974 1667 y Fi(\))1012 1626 y Fl(n)1075 1573 y Fn(X)1085 1785 y Fl(q)r Fk(6)p Fm(=)p Fl(p)1236 1573 y Fn(Y)1239 1785 y Fl(j)t Fk(6)p Fm(=)p Fl(q)1380 1573 y Fn(Y)1724 1600 y Fi(4)p Fj(\031)1832 1564 y Fm(3)p 1533 1644 V 1533 1736 a Fi(sinh\()p Fj(\013)23 b Fo(\000)g Fj(\025)1987 1751 y Fl(j)2023 1736 y Fi(\))2071 1667 y Fj(e)2116 1626 y Fm(\()p Fl(n)p Fk(\000)p Fm(1)p Fk(\000)p Fm(2)p Fl(m)p Fm(\))p Fl(\025)2496 1634 y Ff(q)2537 1667 y Fj(\037)p Fi(\()p Fj(\025)2693 1682 y Fl(p)2754 1667 y Fo(\000)g Fj(\025)2911 1682 y Fl(q)2949 1667 y Fi(\)\()p Fj(\025)3082 1626 y Fl(l)3082 1692 y(p)3143 1667 y Fo(\000)g Fj(\025)3300 1626 y Fl(l)3300 1692 y(q)3338 1667 y Fi(\))234 1990 y Fy(Returning)h(to)h Fj(I)811 1942 y Fl(p)803 2017 y(k)r(;m)927 1990 y Fy(:)710 2268 y Fj(I)761 2221 y Fl(p)753 2296 y(k)r(;m)905 2268 y Fi(=)j(4)p Fj(\031)t(i)p Fi(\(4)p Fj(\031)1296 2227 y Fm(3)1335 2268 y Fi(\))1373 2227 y Fl(n)1420 2268 y Fi(\()p Fo(\000)p Fi(1\))1622 2227 y Fl(k)1681 2173 y Fn(X)1691 2385 y Fl(q)r Fk(6)p Fm(=)p Fl(p)1842 2173 y Fn(Y)1845 2385 y Fl(j)t Fk(6)p Fm(=)p Fl(q)2251 2200 y Fi(1)p 1995 2245 560 4 v 1995 2336 a(sinh\()p Fj(\025)2264 2351 y Fl(q)2324 2336 y Fo(\000)23 b Fj(\025)2481 2351 y Fl(j)2518 2336 y Fi(\))2565 2268 y Fj(e)2610 2227 y Fm(\()p Fl(n)p Fk(\000)p Fm(1)p Fk(\000)p Fm(2)p Fl(m)p Fm(\))p Fl(\025)2990 2235 y Ff(q)3031 2268 y Fj(\037)p Fi(\()p Fj(\025)3187 2283 y Fl(p)3249 2268 y Fo(\000)f Fj(\025)3405 2283 y Fl(q)3443 2268 y Fi(\))900 2540 y Fo(\002)g Fi(\()p Fj(\033)1092 2555 y Fl(k)1135 2540 y Fi(\()p Fj(\025)1230 2555 y Fm(1)1270 2540 y Fj(;)17 b Fo(\001)g(\001)g(\001)31 b Fj(;)1527 2513 y Fn(b)1507 2540 y Fj(\025)1564 2555 y Fl(p)1603 2540 y Fj(;)17 b Fo(\001)g(\001)g(\001)31 b Fj(;)17 b(\025)1897 2555 y Fl(n)1944 2540 y Fi(\))22 b 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y(more)24 b(identity)g(which)g(replaces)h(Riemann)g(bilinear)f(relation:)1445 570 y Fl(n)p Fk(\000)p Fm(1)1440 600 y Fn(X)1447 812 y Fl(k)r Fm(=1)1600 695 y Fi(\()p Fj(A)1711 710 y Fk(\000)p Fl(k)r(;p)1864 695 y Fj(A)1937 710 y Fl(k)r(;q)2055 695 y Fo(\000)f Fj(A)2228 710 y Fk(\000)p Fl(k)r(;q)2379 695 y Fj(A)2452 710 y Fl(k)r(;p)2550 695 y Fi(\))28 b(=)f(0)1169 b Fy(\(40\))234 977 y(Calculating)24 b(minors)g(of)h(the)f(matrix)g Fj(B)30 b Fy(one)25 b(\002nds:)655 1249 y(det)p Fi(\()827 1224 y(~)815 1249 y Fj(I)858 1264 y Fl(j)887 1276 y Ff(l)910 1264 y Fl(;m)997 1249 y Fi(\))p Fo(j)1063 1264 y Fl(l)q(;m)p Fm(=1)p Fl(;)p Fk(\001\001\001)n Fl(n)p Fk(\000)p Fm(1)1499 1249 y Fi(=)j(2)1662 1174 y Fg(\()p Ff(n)p Fe(\000)p Fg(1\)\()p Ff(n)p Fe(\000)p Fg(2\))p 1662 1193 330 3 v 1811 1234 a(2)2005 1249 y Fj(A)2078 1264 y Fl(j)2107 1273 y Fg(1)2142 1264 y Fl(;)p Fk(\001\001\001)10 b Fl(;j)2281 1276 y Ff(l)2367 1124 y Fl(n)2325 1154 y Fn(Y)2326 1363 y Fl(p)p Fm(=1)2496 1181 y Fi(1)p 2479 1226 83 4 v 2479 1317 a Fj(\016)2522 1332 y Fl(p)2581 1107 y Fn(P)2687 1181 y Fi(\()p Fj(\016)2768 1196 y Fl(p)2807 1181 y Fj(e)2852 1145 y Fl(\025)2893 1153 y Ff(p)2934 1181 y Fi(\))17 b(exp)q(\()p Fo(\000)3270 1107 y Fn(P)3391 1181 y Fj(\025)3448 1196 y Fl(p)3488 1181 y Fi(\))p 2581 1226 945 4 v 2723 1242 a Fn(Q)2715 1403 y Fl(i)2201 4725 y Fn(h)2258 4768 y Fj(n)p 2258 4812 59 4 v 2263 4904 a Fi(2)2326 4725 y Fn(i)88 5059 y Fy(do)c(not)g(contrib)n(ute)g(to)h(the)f(\002nal) h(result.)234 5179 y(No)n(w)f(we)h(need)g(the)g(follo)n(wing)88 5299 y Fa(Assumption.)31 b Fy(If)25 b Fj(l)r Fi(\()p Fo(f)p Fj(j)6 b Fo(g)p Fi(\))27 b Fo(\024)1129 5219 y Fn(\002)1181 5260 y Fl(n)p 1181 5276 43 4 v 1185 5334 a Fm(2)1234 5219 y Fn(\003)1300 5299 y Fy(then)956 5554 y Fj(P)1033 5503 y Fk(f)p Fl(\017)p Fk(g)1019 5585 y(f)p Fl(j)t Fk(g)1136 5554 y Fi(\()p Fj(\025)1231 5569 y Fm(1)1293 5554 y Fo(\000)1402 5486 y Fj(\031)t(i)p 1402 5531 93 4 v 1424 5622 a Fi(2)1526 5554 y(+)22 b Fj(i\016)1700 5569 y Fm(1)1740 5554 y Fj(;)17 b Fo(\001)g(\001)g(\001)32 b Fj(;)17 b(\025)2035 5569 y Fm(1)2096 5554 y Fi(+)2204 5486 y Fj(\031)t(i)p 2204 5531 V 2225 5622 a Fi(2)2328 5554 y Fo(\000)23 b Fj(i\016)2504 5569 y Fm(1)2544 5554 y Fi(\))k(=)h Fo(O)s Fi(\()p Fj(\016)2880 5513 y Fl(n)p Fk(\000)p Fm(2)p Fl(l)q Fm(\()p Fk(f)p Fl(j)t Fk(g)p Fm(\))3196 5554 y Fi(\))703 b Fy(\(44\))2046 5871 y(14)p eop %%Page: 15 15 15 14 bop 234 289 a Fy(Unfortunately)-6 b(,)21 b(for)i(the)f(moment)f (we)h(are)h(not)f(able)g(to)g(pro)o(v)o(e)g(it.)29 b(Let)22 b(us)g(mention)f(at)h(least)g(some)g(ar)n(guments)88 409 y(why)34 b(we)h(belie)n(v)o(e)f(in)h(this)f(Assumption?)60 b(First,)37 b(suppose)d(the)h(de)o(gree)g(in)g Fj(\016)k Fy(is)c(smaller)f(then)h(in)g(\(44\).)62 b(Then)88 529 y(some)32 b(huge)h(cancellation)g(must)f(tak)o(e)h(place)h(because)f (all)g(contrib)n(utions)e(from)i(leading)g(term)g(of)g(asymptotic)88 650 y(described)27 b(by)h Fj(A)698 665 y Fk(f)p Fl(j)t Fk(g)833 650 y Fy(are)g(singular)f(in)g(the)h(limit.)38 b(The)28 b(estimation)e(\(44\))h(is)h(v)n(alid)e(for)i Fj(n)34 b Fi(=)f(2)p Fj(;)17 b Fi(3)p Fy(,)28 b(these)f(cases)h(we)88 770 y(were)g(able)g(to)f(calculate)h(by)f(Maple.)39 b(Certainly)-6 b(,)28 b(the)f(best)h(w)o(ay)f(to)h(pro)o(v)o(e)e(the)i(Assumption)d (is)i(to)h(calculate)g(the)88 901 y(polynomials)d Fj(P)685 850 y Fk(f)p Fl(\017)p Fk(g)671 932 y(f)p Fl(j)t Fk(g)788 901 y Fy(.)40 b(These)27 b(polynomials)f(are)i(related)g(to)g(the)f (polynomials)f Fj(K)2946 916 y Fl(j)2975 925 y Fg(1)3009 916 y Fl(;)p Fk(\001\001\001)10 b Fl(;j)3148 925 y Ff(n)p Fe(\000)p Fg(1)3300 901 y Fy(through)27 b(the)h(formula)88 1021 y(\(34\).)j(The)24 b(latter)h(polynomials)e(are)i(gi)n(v)o(en)f (by)g(the)h(formulae)g(\(14\),)g(\(25\))f(and)h(\(26\).)31 b(The)25 b(main)f(problem)g(here)i(is)88 1142 y(to)h(tak)o(e)h(the)g (limit)e Fj(\016)795 1157 y Fl(k)872 1142 y Fo(!)33 b Fi(0)28 b Fy(for)g(the)g(inte)o(grals)e(\(25\))i(when)g(the)g(spectral) g(parameters)g Fj(\014)3191 1157 y Fl(k)3262 1142 y Fy(are)g(tak)o(en)g (in)g(the)g(form)88 1262 y(\(35\).)i(W)-8 b(e)26 b(lea)n(v)o(e)e(this)g (question)g(for)h(further)g(publication.)234 1382 y(Let)e(us)g(suppose) g(that)g(the)g(Assumption)e(is)i(true.)30 b(Then)23 b(still)f (important)g(cancellations)h(must)f(occur)i(because)88 1503 y(only)d(one)i(kind)f(of)g(terms)g(of)h(the)g(order)g Fj(\016)1534 1467 y Fl(n)1603 1503 y Fy(must)f(remain)g(in)g(the)h(T)-8 b(ailor)22 b(series:)29 b(that)22 b(with)3278 1428 y Fn(Q)3389 1503 y Fj(\016)3432 1518 y Fl(p)3472 1503 y Fy(.)30 b(The)22 b(result)g(will)88 1623 y(look)i(lik)o(e:)641 1865 y Fj(g)t Fi(\()p Fj(\025)787 1880 y Fm(1)848 1865 y Fo(\000)957 1798 y Fj(\031)t(i)p 957 1842 93 4 v 979 1933 a Fi(2)1059 1865 y Fj(;)17 b Fo(\001)g(\001)g(\001)31 b Fj(;)17 b(\025)1353 1880 y Fm(1)1415 1865 y Fi(+)1523 1798 y Fj(\031)t(i)p 1523 1842 V 1544 1933 a Fi(2)1625 1865 y(\))1663 1824 y Fl(\017)1692 1833 y Fg(1)1726 1824 y Fl(;)p Fk(\001\001\001)10 b Fl(;\017)1865 1833 y Fg(2)p Ff(n)1969 1865 y 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3033 y Fg(1)2829 3024 y Fl(;)p Fk(\001\001\001)10 b Fl(;\017)2968 3033 y Fg(2)p Ff(n)2766 3099 y Fl(k)2803 3108 y Fg(1)2837 3099 y Fl(;)p Fk(\001\001\001)g Fl(;k)2984 3108 y Fg(2)p Ff(m)3077 3072 y Fi(\()p Fj(\025)3172 3087 y Fm(1)3211 3072 y Fj(;)17 b Fo(\001)g(\001)g(\001)31 b Fj(;)17 b(\025)3505 3087 y Fl(n)3552 3072 y Fi(\))88 3363 y Fy(where)381 3338 y Fi(~)355 3363 y Fj(A)428 3316 y Fl(\017)457 3325 y Fg(1)491 3316 y Fl(;)p Fk(\001\001\001)10 b Fl(;\017)630 3325 y Fg(2)p Ff(n)428 3391 y Fl(k)465 3400 y Fg(1)500 3391 y Fl(;)p Fk(\001\001\001)g Fl(;k)647 3400 y Fg(2)p Ff(m)763 3363 y Fy(are)25 b(some)e(rational)h(functions.)29 b(Doing)23 b(a)i(substitution)c(lik)o(e)i Fj(\025)2958 3378 y Fl(k)3029 3363 y Fo(!)k Fj(\031)t Fi(\()p Fj(z)3298 3378 y Fl(k)3360 3363 y Fo(\000)3472 3324 y Fl(i)p 3466 3341 36 4 v 3466 3398 a Fm(2)3512 3363 y Fi(\))d Fy(and)g(using)f(the) 88 3484 y(formula)h(\(30\))h(we)g(arri)n(v)o(e)f(at)h(the)g(formula)172 3835 y Fo(h)253 3710 y Fl(n)211 3740 y Fn(Y)213 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Fj(z)2886 3850 y Fl(k)2923 3859 y Fg(2)2962 3835 y Fi(\))17 b Fo(\001)g(\001)g(\001)e Fj(G)p Fi(\()p Fj(z)3310 3850 y Fl(k)3347 3859 y Fg(2)p Ff(m)p Fe(\000)p Fg(1)3540 3835 y Fo(\000)23 b Fj(z)3685 3850 y Fl(k)3722 3859 y Fg(2)p Ff(m)3815 3835 y Fi(\))84 b Fy(\(46\))88 4161 y(where)25 b Fj(A)429 4113 y Fl(\017)458 4122 y Fg(1)492 4113 y Fl(;)p Fk(\001\001\001)10 b Fl(;\017)631 4122 y Fg(2)p Ff(n)429 4188 y Fl(k)466 4197 y Fg(1)500 4188 y Fl(;)p Fk(\001\001\001)g Fl(;k)647 4197 y Fg(2)p Ff(m)740 4161 y Fi(\()p Fj(z)823 4176 y Fm(1)863 4161 y Fj(;)17 b Fo(\001)g(\001)g(\001)31 b Fj(;)17 b(z)1145 4176 y Fl(n)1192 4161 y Fi(\))25 b Fy(are)g(rational)g(functions)e (related)i(in)g(an)g(ob)o(vious)e(w)o(ay)h(to)h(the)f(functions)113 4276 y Fi(~)88 4301 y Fj(A)161 4254 y Fl(\017)190 4263 y Fg(1)224 4254 y Fl(;)p Fk(\001\001\001)10 b Fl(;\017)363 4263 y Fg(2)p Ff(n)161 4329 y Fl(k)198 4338 y Fg(1)232 4329 y Fl(;)p Fk(\001\001\001)g Fl(;k)379 4338 y Fg(2)p Ff(m)472 4301 y Fi(\()p Fj(\025)567 4316 y Fm(1)606 4301 y Fj(;)17 b Fo(\001)g(\001)g(\001)31 b Fj(;)17 b(\025)900 4316 y Fl(n)947 4301 y Fi(\))39 b Fy(and)g(the)h(function)e Fj(G)i Fy(is)e(de\002ned)i(by)f(the)h(formula)e(\(3.6\))i(of)f(the)g(w) o(ork)h([12)o(].)75 b(W)-8 b(e)88 4421 y(see)34 b(w)o(onderful)f (agreement)h(with)f(the)h(ansatz)g(\(3.20\))g(from)f(the)h(abo)o(v)o(e) f(paper)h(for)g(the)g(emptiness)e(formation)88 4542 y(probability)23 b Fj(P)613 4557 y Fl(n)660 4542 y Fi(\()p Fj(z)743 4557 y Fm(1)782 4542 y Fj(;)17 b Fo(\001)g(\001)g(\001)31 b Fj(;)17 b(z)1064 4557 y Fl(n)1111 4542 y Fi(\))25 b Fy(which)g(w)o(as)f(based)h(on)g(quite)f(dif)n(ferent)g(ar)n(guments.) 88 4881 y Fp(4)143 b(Conclusion)88 5105 y Fy(In)39 b(this)f(paper)h(we) h(ha)n(v)o(e)e(applied)h(a)g(ne)n(w)g(form)g(of)g(solution)e(to)i(the)g (qKZ)g(on)f(le)n(v)o(el)g(-4)h(for)h(the)f(correlation)88 5225 y(functions)26 b(of)i(the)f(XXX)h(model.)38 b(The)o(y)27 b(are)h(gi)n(v)o(en)e(by)i(the)f(formula)g(\(46\))h(in)f(a)h(full)f (agreement)h(with)f(the)g(result)88 5345 y(of)32 b(paper)g([12].)52 b(The)32 b(mathematical)f(reasons)g(for)h(a)h(reducibility)d(of)i (solutions)e(of)i(the)g(qKZ)f(on)h(le)n(v)o(el)e(-4)i(and)88 5466 y(the)c(correlation)g(functions)f(to)h(one-dimensional)e(inte)o (grals)h(is)h(the)h(special)f(kind)f(of)i(the)f(cohomologies)e(of)j (the)88 5586 y(deformed)h(Jacobi)g(v)n(arieties.)46 b(Our)31 b(pre)n(vious)e(conjecture)h(that)g(in)g(the)h(homogeneous)d(limit)h Fj(z)3482 5601 y Fl(j)3557 5586 y Fo(!)38 b Fi(0)30 b Fy(both)g(the)2046 5871 y(15)p eop %%Page: 16 16 16 15 bop 88 289 a Fy(emptiness)35 b(formation)g(probability)g(and)h (other)h(correlation)f(functions)f(may)h(be)h(e)o(xpressed)f(in)g (terms)g(of)h(the)88 409 y(Riemann)28 b(zeta)i(function)e(at)h(odd)f (ar)n(guments)g(and)h(rational)g(coef)n(\002cients)g([10)o(,)g(11])g (follo)n(ws)e(from)i(the)g(formula)88 529 y(\(46\))24 b(and)h(the)g(e)o(xpansion)e(\(3.13\))i(of)g(the)f(paper)i([12)o(].)234 650 y(Unfortunately)-6 b(,)20 b(there)i(are)g(still)e(some)h(technical) g(problems)f(that)h(were)h(not)f(solv)o(ed)f(or)i(solv)o(ed)e(only)g (partially)88 770 y(in)30 b(this)g(paper)-5 b(.)49 b(First)30 b(we)i(are)f(not)f(completely)g(satis\002ed)h(with)f(the)h(formulae)f (\(14\),)i(\(25\))f(and)g(\(26\))g(which)g(are)88 891 y(necessary)d(for)h(de\002ning)e(the)h(polynomials)e Fj(K)1756 906 y Fl(j)1785 915 y Fg(1)1820 906 y Fl(;)p Fk(\001\001\001)10 b Fl(;j)1959 915 y Ff(n)p Fe(\000)p Fg(1)2083 891 y Fy(.)41 b(In)28 b(particular)l(,)h(we)f(ha)n(v)o(e)g (not)g(succeeded)h(in)e(\002nding)h(a)88 1011 y(general)21 b(formula)f(for)i(the)f(functions)e Fj(f)1458 1026 y Fl(r)1518 1011 y Fy(from)h(the)h(formula)g(\(28\).)29 b(W)-8 b(e)22 b(ha)n(v)o(e)f(done)f(it)h(completely)f(for)h Fj(n)28 b Fo(\024)g Fi(6)21 b Fy(\(see)88 1131 y(Appendix\).)32 b(F)o(or)26 b(a)g(generic)g(case)g(we)g(ha)n(v)o(e)f(only)g(formulated) g(some)g(general)h(conditions)e(for)i(these)g(functions.)88 1252 y(W)-8 b(e)25 b(may)f(hope)h(that,)f(probably)-6 b(,)24 b(some)g(more)g(e)o(xplicit)g(formulae)g(should)g(e)o(xist)f (for)j(those)e(polynomials.)234 1372 y(Also)29 b(the)h(challenging)f (problem)g(is)g(to)g(de\002ne)i(the)e(rational)h(functions)f Fj(A)2898 1336 y Fl(k)2935 1345 y Fg(1)2969 1336 y Fl(;)p Fk(\001\001\001)10 b Fl(;k)3116 1345 y Fg(2)p Ff(m)2898 1397 y Fl(\017)2927 1406 y Fg(1)2961 1397 y Fl(;)p Fk(\001\001\001)g Fl(;\017)3100 1406 y Fg(2)p Ff(n)3209 1372 y Fi(\()p Fj(z)3292 1387 y Fm(1)3331 1372 y Fj(;)17 b Fo(\001)g(\001)g(\001)32 b Fj(;)17 b(z)3614 1387 y Fl(n)3661 1372 y Fi(\))29 b Fy(e)o(xplicitly)88 1493 y(using)23 b(the)i(results)f(of)h(the)g (Section)f(2.)234 1613 y(W)-8 b(e)25 b(hope)f(to)g(do)g(it)g(in)g(our)g (further)h(publication.)k(W)-8 b(e)24 b(also)g(hope)g(to)g(generalize)h (all)f(the)h(abo)o(v)o(e)e(results)h(to)g(the)88 1733 y(case)h(of)g(the)f(XXZ)h(spin)f(chain.)88 2072 y Fp(5)143 b(Ackno)o(wledgements)88 2296 y Fy(The)30 b(authors)f(w)o(ould)h(lik)o 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5554 y Fm(6)1604 5539 y Fi(\))1642 5554 y Fk(\000\000\000)p Fm(+++)2003 5539 y Fi(=)2199 5471 y Fj(g)t Fi(\()p Fj(\033)2343 5486 y Fm(1)2382 5471 y Fj(;)g(\033)2481 5486 y Fm(2)2521 5471 y Fi(\))p 2116 5516 526 4 v 2116 5607 a Fj(\033)2171 5622 y Fm(2)2233 5607 y Fo(\000)23 b Fj(\033)2388 5622 y Fm(1)2450 5607 y Fo(\000)g Fj(\031)t(i)3937 5539 y Fy(\(48\))2046 5871 y(16)p eop %%Page: 17 17 17 16 bop 88 289 a Fy(where)38 b Fj(g)k Fy(is)37 b(a)i(polynomial)d (for)i(which)g(the)g(dependence)g(on)g(the)g(rapidities)f Fj(\014)3003 304 y Fm(1)3043 289 y Fj(;)17 b(:)g(:)g(:)e(;)i(\014)3316 304 y Fm(6)3394 289 y Fy(is)37 b(implied.)70 b(Then)88 409 y(according)24 b(to)h(the)f(formula)h(\(22\))843 681 y Fj(h)899 696 y Fm(0)939 681 y Fi(\()p Fj(\033)1032 696 y Fm(1)1072 681 y Fj(;)17 b(\033)1171 696 y Fm(2)1210 681 y Fi(\))28 b(=)1472 613 y Fj(g)t Fi(\()p Fj(\033)1616 628 y Fm(1)1655 613 y Fj(;)17 b(\033)1754 628 y Fm(2)1794 613 y Fi(\))p 1389 658 526 4 v 1389 749 a Fj(\033)1444 764 y Fm(2)1506 749 y Fo(\000)23 b Fj(\033)1661 764 y Fm(1)1723 749 y Fo(\000)g Fj(\031)t(i)1947 681 y Fo(\000)2139 613 y Fj(g)t Fi(\()p Fj(\033)2283 628 y Fm(2)2323 613 y Fj(;)17 b(\033)2422 628 y Fm(1)2461 613 y Fi(\))p 2057 658 V 2057 749 a Fj(\033)2112 764 y Fm(1)2174 749 y Fo(\000)22 b Fj(\033)2328 764 y Fm(2)2390 749 y Fo(\000)h Fj(\031)t(i)3937 681 y Fy(\(49\))88 940 y(The)h(formula)h(\(26\))g(contains)f(tw)o(o)g (terms)g(for)h Fj(p)j Fi(=)f(0)e Fy(and)g Fj(p)j Fi(=)f(1)e Fy(respecti)n(v)o(ely:)1166 1160 y Fo(L)p Fi(\()p Fj(p)1322 1175 y Fm(1)1361 1160 y Fj(;)17 b(p)1454 1175 y Fm(2)1493 1160 y Fi(\))27 b(=)h Fj(I)1705 1175 y Fm(0)1767 1160 y Fi(+)22 b Fj(I)1908 1175 y Fm(1)3937 1160 y Fy(\(50\))88 1380 y(with)36 b(polynomials)g Fj(p)882 1395 y Fm(1)959 1380 y Fy(and)i Fj(p)1190 1395 y Fm(2)1267 1380 y Fy(which)f(belong)g (to)g(the)h(set)f(of)h(polynomials)d Fj(s)2977 1395 y Fk(\000)p Fm(2)3071 1380 y Fj(;)17 b(s)3161 1395 y Fk(\000)p Fm(1)3255 1380 y Fj(;)g(s)3345 1395 y Fm(1)3384 1380 y Fj(;)g(s)3474 1395 y Fm(2)3514 1380 y Fy(.)69 b(The)37 b(term)h Fj(I)4064 1395 y Fm(0)88 1500 y Fy(corresponds)24 b(to)g(the)h(case)g(when)g(partitions)1614 1720 y Fj(S)34 b Fi(=)p 1811 1640 66 4 v 27 w Fj(S)g Fi(=)27 b Fo(;)100 b Fj(T)41 b Fi(=)28 b(\(1)p Fj(;)17 b Fi(2\))501 2089 y Fj(I)544 2104 y Fm(0)611 2089 y Fo(\021)28 b Fj(I)759 2105 y Fk(;)p Fl(;)p Fk(;)p Fl(;)p Fm(\(1)p Fl(;)p Fm(2\))1046 2089 y Fi(=)1150 1954 y Fn(Z)1205 2179 y Fm(\000)1249 2188 y Fg(1)1314 2022 y Fj(d\033)1420 2037 y Fm(1)p 1314 2066 146 4 v 1316 2158 a Fi(2)p Fj(\031)t(i)1486 1954 y Fn(Z)1542 2179 y Fm(\000)1586 2188 y Fg(2)1651 2022 y Fj(d\033)1757 2037 y Fm(2)p 1651 2066 V 1653 2158 a Fi(2)p Fj(\031)t(i)1807 2089 y(X)1888 2104 y Fm(1)1927 2089 y Fi(\()p Fj(\033)2020 2104 y Fm(1)2060 2089 y Fi(\)\010\()p Fj(\033)2261 2104 y Fm(1)2301 2089 y Fi(\))p Fj(X)2420 2104 y Fm(2)2459 2089 y Fi(\()p Fj(\033)2552 2104 y Fm(2)2592 2089 y Fi(\)\010\()p Fj(\033)2793 2104 y Fm(2)2833 2089 y Fi(\))p Fj(h)2927 2104 y 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2112 5517 a Fm(2)2173 5483 y Fi(\))3937 5411 y Fy(\(56\))2046 5871 y(17)p eop %%Page: 18 18 18 17 bop 234 289 a Fy(In)25 b(order)g(to)f(compensate)g(this)g (additional)g(term)g Fj(J)2028 304 y Fm(0)2092 289 y Fy(we)h(need)g(the)g(second)f(term)h Fj(I)3145 304 y Fm(1)3209 289 y Fy(in)g(the)f(formula)h(\(50\).)30 b(In)88 409 y(this)23 b(case)j(there)f(is)f(only)g(one)h(partition)f (corresponding)f(to)i Fj(p)i Fi(=)h(1)d Fy(which)f(satis\002es)g(the)h (conditions)e(\(24\))1548 629 y Fj(S)34 b Fi(=)27 b(\(1\))p 1970 549 66 4 v 100 w Fj(S)33 b Fi(=)28 b(\(2\))99 b Fj(T)41 b Fi(=)28 b Fo(;)751 998 y Fj(I)794 1013 y Fm(1)861 998 y Fo(\021)g Fj(I)1009 1014 y Fm(\(1\))p Fl(;)p Fm(\(2\))p Fl(;)p Fk(;)1296 998 y Fi(=)1399 863 y Fn(Z)1454 1088 y Fm(\000)1498 1097 y Fg(1)1564 931 y Fj(d\033)1670 946 y Fm(1)p 1564 975 146 4 v 1566 1066 a Fi(2)p Fj(\031)t(i)1719 998 y(X)1800 1013 y Fm(1)1840 998 y Fi(\()p Fj(\033)1933 1013 y Fm(1)1973 998 y Fi(\))p Fj(p)2060 1013 y Fm(2)2099 998 y Fi(\()p 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2057 a Fj(P)14 b Fi(\()p Fj(\033)2390 2072 y Fm(1)2452 2057 y Fi(+)2560 2018 y Fl(\031)r(i)p 2560 2034 67 4 v 2576 2091 a Fm(2)2637 2057 y Fi(\))88 2282 y Fy(One)29 b(can)g(check)h(that)f(due)g(to)g(the)g(formula)f(\(23\))i (the)f(polynomial)e Fj(g)t Fi(\()p Fj(\033)2640 2297 y Fm(1)2679 2282 y Fj(;)17 b(\033)2778 2297 y Fm(1)2843 2282 y Fi(+)25 b Fj(\031)t(i)p Fi(\))30 b Fy(has)f(a)g(di)n(visor)f Fj(P)14 b Fi(\()p Fj(\033)3813 2297 y Fm(1)3877 2282 y Fi(+)3988 2243 y Fl(\031)r(i)p 3988 2259 V 4004 2317 a Fm(2)4065 2282 y Fi(\))88 2403 y Fy(i.e.)44 b Fi([)p Fj(R)355 2418 y Fm(1)395 2403 y Fj(h)451 2418 y Fm(0)490 2403 y Fi(\()p Fj(\033)583 2418 y Fm(1)623 2403 y Fj(;)17 b(\033)722 2418 y Fm(2)761 2403 y Fi(\)])30 b Fy(is)f(a)g(polynomial.) 43 b(Therefore)30 b(the)f(operator)g Fj(\016)2530 2367 y Fk(\000)p Fm(1)2526 2427 y Fl(\033)2566 2436 y Fg(1)2654 2403 y Fy(is)g(well-de\002ned)g(in)g(this)g(case)g(and)h(may)88 2523 y(be)25 b(calculated)f(e)o(xplicitly)-6 b(.)234 2644 y(In)32 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Fo(\000)e Fj(I)2163 4149 y Fm(1)2203 4153 y Fk(j)2223 4162 y Fg(1)p Fe($)p Fg(2)88 4309 y Fy(and)i(we)h(come)g(to)g(the)f(result)g(that)h(the)f(r)-5 b(.h.s.)30 b(of)25 b(the)g(formula)f(\(50\))h(is)f(the)h(sk)o(e)n (w-symmetric)e(function.)234 4429 y(3.)30 b Fa(The)c(case)f Fj(n)j Fi(=)g(4)234 4550 y Fy(First)d(let)f(us)h(represent)g(the)f (function)g Fj(D)k Fy(from)c(\(23\))h(in)g(analogous)f(to)g(\(48\))h (form:)818 4770 y Fj(D)s Fi(\()p Fj(\033)995 4785 y Fm(1)1035 4770 y Fj(;)17 b(\033)1134 4785 y Fm(2)1173 4770 y Fj(;)g(\033)1272 4785 y Fm(3)1312 4770 y Fo(j)p Fj(\014)1395 4785 y Fm(1)1434 4770 y Fj(;)g(:)g(:)g(:)f(;)h(\014)1708 4785 y Fm(8)1747 4770 y Fi(\))1785 4785 y Fk(\000\000\000\000)p Fm(++++)2256 4770 y Fi(=)1481 4914 y Fj(g)t Fi(\()p Fj(\033)1625 4929 y Fm(1)1664 4914 y Fj(;)g(\033)1763 4929 y Fm(2)1803 4914 y Fj(;)g(\033)1902 4929 y Fm(3)1941 4914 y Fi(\))p 828 4958 1805 4 v 828 5049 a(\()p Fj(\033)921 5064 y Fm(2)983 5049 y Fo(\000)23 b Fj(\033)1138 5064 y Fm(1)1200 5049 y Fo(\000)f Fj(\031)t(i)p Fi(\)\()p Fj(\033)1522 5064 y Fm(3)1584 5049 y Fo(\000)h Fj(\033)1739 5064 y Fm(1)1801 5049 y Fo(\000)g Fj(\031)t(i)p Fi(\)\()p Fj(\033)2124 5064 y Fm(3)2186 5049 y Fo(\000)f Fj(\033)2340 5064 y Fm(2)2402 5049 y Fo(\000)h Fj(\031)t(i)p Fi(\))3937 4981 y Fy(\(60\))88 5251 y(where)i Fj(g)j Fy(is)c(a)h(polynomial)e(of)i (both)f(v)n(ariables)g Fj(\033)1817 5266 y Fm(1)1857 5251 y Fj(;)17 b(\033)1956 5266 y Fm(2)1995 5251 y Fj(;)g(\033)2094 5266 y Fm(3)2159 5251 y Fy(and)24 b(of)h(the)g(rapidities)f Fj(\014)3033 5266 y Fm(1)3072 5251 y Fj(;)17 b(:)g(:)g(:)f(;)h(\014) 3346 5266 y Fm(8)3385 5251 y Fy(.)31 b(An)24 b(e)o(xplicit)g(form)88 5372 y(of)j(this)g(polynomial)f(is)h(de\002ned)h(by)f(the)h(formula)f (\(23\).)39 b(F)o(or)28 b(us)f(it)g(is)g(important)g(that)g(this)f (polynomial)g(has)i(the)88 5492 y(property)c(that)g(three)h (polynomials)88 5612 y Fj(g)t Fi(\()p Fj(\033)232 5627 y Fm(1)271 5612 y Fj(;)17 b(\033)370 5627 y Fm(1)431 5612 y Fi(+)22 b Fj(\031)t(i;)17 b(\033)720 5627 y Fm(3)760 5612 y Fi(\))p Fj(;)g(g)t Fi(\()p Fj(\033)986 5627 y Fm(1)1025 5612 y Fj(;)g(\033)1124 5627 y Fm(3)1163 5612 y Fj(;)g(\033)1262 5627 y Fm(1)1324 5612 y Fi(+)22 b Fj(\031)t(i)p Fi(\))p Fj(;)17 b(g)t Fi(\()p Fj(\033)1740 5627 y Fm(3)1779 5612 y Fj(;)g(\033)1878 5627 y Fm(1)1918 5612 y Fj(;)g(\033)2017 5627 y Fm(1)2078 5612 y Fi(+)22 b Fj(\031)t(i)p Fi(\))k Fy(contain)e(di)n(visor)2046 5871 y(18)p eop %%Page: 19 19 19 18 bop 88 289 a Fj(P)14 b Fi(\()p Fj(\033)258 304 y Fm(1)319 289 y Fi(+)427 249 y Fl(\031)r(i)p 427 266 67 4 v 443 323 a Fm(2)504 289 y Fi(\))25 b Fy(while)f(the)h(polynomial) d Fj(g)t Fi(\()p Fj(\033)1582 304 y Fm(1)1621 289 y Fj(;)17 b(\033)1720 304 y Fm(1)1782 289 y Fi(+)22 b Fj(\031)t(i;)17 b(\033)2071 304 y Fm(1)2133 289 y Fi(+)22 b(2)p Fj(\031)t(i)p Fi(\))j Fy(is)f(di)n(visible)f(by)88 409 y Fj(P)14 b Fi(\()p Fj(\033)258 424 y Fm(1)319 409 y Fi(+)427 370 y Fl(\031)r(i)p 427 386 V 443 443 a Fm(2)504 409 y Fi(\))p Fj(P)g Fi(\()p Fj(\033)712 424 y Fm(1)773 409 y Fi(+)881 370 y Fm(3)p Fl(\031)r(i)p 881 386 103 4 v 915 443 a Fm(2)993 409 y Fi(\))p Fy(.)31 b(Then)25 b(according)f(to)h(the)f (formula)h(\(22\))507 749 y Fj(h)563 764 y Fm(0)603 749 y Fi(\()p Fj(\033)696 764 y Fm(1)735 749 y Fj(;)17 b(\033)834 764 y Fm(2)874 749 y Fj(;)g(\033)973 764 y Fm(3)1013 749 y Fi(\))27 b(=)898 894 y Fj(g)t Fi(\()p Fj(\033)1042 909 y Fm(1)1081 894 y Fj(;)17 b(\033)1180 909 y Fm(2)1220 894 y Fj(;)g(\033)1319 909 y Fm(3)1358 894 y Fi(\))p 517 938 1260 4 v 517 1029 a(\()p Fj(\033)610 1044 y Fm(21)707 1029 y Fo(\000)23 b Fj(\031)t(i)p Fi(\)\()p Fj(\033)1030 1044 y Fm(31)1127 1029 y Fo(\000)g Fj(\031)t(i)p Fi(\)\()p Fj(\033)1450 1044 y Fm(32)1547 1029 y Fo(\000)g Fj(\031)t(i)p Fi(\))1809 961 y Fo(\000)2300 894 y Fj(g)t Fi(\()p Fj(\033)2444 909 y Fm(2)2483 894 y Fj(;)17 b(\033)2582 909 y Fm(1)2621 894 y Fj(;)g(\033)2720 909 y Fm(3)2760 894 y Fi(\))p 1919 938 V 1919 1029 a(\()p Fj(\033)2012 1044 y Fm(12)2109 1029 y Fo(\000)22 b Fj(\031)t(i)p Fi(\)\()p Fj(\033)2431 1044 y Fm(31)2529 1029 y Fo(\000)g Fj(\031)t(i)p Fi(\)\()p Fj(\033)2851 1044 y Fm(32)2949 1029 y Fo(\000)g Fj(\031)t(i)p Fi(\))3189 961 y(+)898 1162 y Fj(g)t Fi(\()p Fj(\033)1042 1177 y Fm(2)1081 1162 y Fj(;)17 b(\033)1180 1177 y Fm(3)1220 1162 y Fj(;)g(\033)1319 1177 y Fm(1)1358 1162 y Fi(\))p 517 1207 V 517 1298 a(\()p Fj(\033)610 1313 y Fm(32)707 1298 y Fo(\000)23 b Fj(\031)t(i)p Fi(\)\()p Fj(\033)1030 1313 y Fm(12)1127 1298 y Fo(\000)g Fj(\031)t(i)p Fi(\)\()p Fj(\033)1450 1313 y Fm(13)1547 1298 y Fo(\000)g Fj(\031)t(i)p Fi(\))1809 1229 y Fo(\000)2300 1162 y Fj(g)t Fi(\()p Fj(\033)2444 1177 y Fm(3)2483 1162 y Fj(;)17 b(\033)2582 1177 y Fm(2)2621 1162 y Fj(;)g(\033)2720 1177 y Fm(1)2760 1162 y Fi(\))p 1919 1207 V 1919 1298 a(\()p Fj(\033)2012 1313 y Fm(23)2109 1298 y Fo(\000)22 b Fj(\031)t(i)p Fi(\)\()p Fj(\033)2431 1313 y Fm(12)2529 1298 y Fo(\000)g Fj(\031)t(i)p Fi(\)\()p Fj(\033)2851 1313 y Fm(13)2949 1298 y Fo(\000)g Fj(\031)t(i)p Fi(\))3189 1229 y(+)898 1431 y Fj(g)t Fi(\()p Fj(\033)1042 1446 y Fm(3)1081 1431 y Fj(;)17 b(\033)1180 1446 y Fm(1)1220 1431 y Fj(;)g(\033)1319 1446 y Fm(2)1358 1431 y Fi(\))p 517 1475 V 517 1566 a(\()p Fj(\033)610 1581 y Fm(13)707 1566 y Fo(\000)23 b Fj(\031)t(i)p Fi(\)\()p Fj(\033)1030 1581 y Fm(23)1127 1566 y Fo(\000)g Fj(\031)t(i)p Fi(\)\()p Fj(\033)1450 1581 y Fm(21)1547 1566 y Fo(\000)g Fj(\031)t(i)p Fi(\))1809 1498 y Fo(\000)2300 1431 y Fj(g)t Fi(\()p Fj(\033)2444 1446 y Fm(1)2483 1431 y Fj(;)17 b(\033)2582 1446 y Fm(3)2621 1431 y Fj(;)g(\033)2720 1446 y Fm(2)2760 1431 y Fi(\))p 1919 1475 V 1919 1566 a(\()p Fj(\033)2012 1581 y Fm(31)2109 1566 y Fo(\000)22 b Fj(\031)t(i)p Fi(\)\()p Fj(\033)2431 1581 y Fm(23)2529 1566 y Fo(\000)g Fj(\031)t(i)p Fi(\)\()p Fj(\033)2851 1581 y Fm(21)2949 1566 y Fo(\000)g Fj(\031)t(i)p Fi(\))3937 1498 y Fy(\(61\))234 1767 y(As)i(in)h(the)g(pre)n(vious)e(case)i(the)g (formula)f(\(26\))h(contains)f(tw)o(o)h(terms)f(for)h Fj(p)j Fi(=)f(0)e Fy(and)g Fj(p)i Fi(=)h(1)c Fy(respecti)n(v)o(ely:) 1122 1987 y Fo(L)p Fi(\()p Fj(p)1278 2002 y Fm(1)1317 1987 y Fj(;)17 b(p)1410 2002 y Fm(2)1449 1987 y Fj(;)g(p)1542 2002 y Fm(3)1581 1987 y Fi(\))28 b(=)f Fj(I)1793 2002 y Fm(0)1855 1987 y Fi(+)22 b Fj(I)1996 2002 y Fm(1)3937 1987 y Fy(\(62\))88 2207 y(with)i(polynomials)e Fj(p)856 2222 y Fm(1)896 2207 y Fj(;)17 b(p)989 2222 y Fm(2)1028 2207 y Fj(;)g(p)1121 2222 y Fm(3)1185 2207 y Fy(from)24 b(the)h(set)f Fj(s)1731 2222 y Fk(\000)p Fm(3)1826 2207 y Fj(;)17 b(s)1916 2222 y Fk(\000)p Fm(2)2010 2207 y Fj(;)g(s)2100 2222 y Fk(\000)p Fm(1)2194 2207 y Fj(;)g(s)2284 2222 y Fm(1)2323 2207 y Fj(;)g(s)2413 2222 y Fm(2)2452 2207 y Fj(;)g(s)2542 2222 y Fm(3)2581 2207 y Fy(.)234 2328 y(The)25 b(term)f Fj(I)664 2343 y Fm(0)729 2328 y Fy(corresponds)g(to)g(the)h(case)g(when)g(partitions)1568 2548 y Fj(S)33 b Fi(=)p 1765 2468 66 4 v 28 w Fj(S)g Fi(=)28 b Fo(;)99 b Fj(T)42 b Fi(=)27 b(\(1)p Fj(;)17 b Fi(2)p Fj(;)g Fi(3\))320 2867 y Fj(I)363 2882 y Fm(0)431 2867 y Fo(\021)28 b Fj(I)579 2883 y Fk(;)p Fl(;)p Fk(;)p Fl(;)p Fm(\(1)p Fl(;)p Fm(2)p Fl(;)p Fm(3\))920 2867 y Fi(=)337 2938 y Fn(Z)392 3163 y Fm(\000)436 3172 y Fg(1)502 3006 y Fj(d\033)608 3021 y Fm(1)p 502 3050 146 4 v 504 3141 a Fi(2)p Fj(\031)t(i)674 2938 y Fn(Z)729 3163 y Fm(\000)773 3172 y Fg(2)839 3006 y Fj(d\033)945 3021 y Fm(2)p 839 3050 V 841 3141 a Fi(2)p Fj(\031)t(i)1011 2938 y Fn(Z)1066 3163 y Fm(\000)1110 3172 y Fg(3)1175 3006 y Fj(d\033)1281 3021 y Fm(3)p 1175 3050 V 1177 3141 a Fi(2)p Fj(\031)t(i)1331 3073 y(X)1412 3088 y Fm(1)1451 3073 y Fi(\()p Fj(\033)1544 3088 y Fm(1)1584 3073 y Fi(\)\010\()p Fj(\033)1785 3088 y Fm(1)1825 3073 y Fi(\))p Fj(X)1944 3088 y Fm(2)1984 3073 y Fi(\()p Fj(\033)2077 3088 y Fm(2)2117 3073 y Fi(\)\010\()p Fj(\033)2318 3088 y Fm(2)2358 3073 y Fi(\))p Fj(X)2477 3088 y Fm(3)2516 3073 y Fi(\()p Fj(\033)2609 3088 y Fm(3)2649 3073 y Fi(\)\010\()p Fj(\033)2850 3088 y Fm(3)2890 3073 y Fi(\))p Fj(h)2984 3088 y Fm(0)3024 3073 y Fi(\()p Fj(\033)3117 3088 y Fm(1)3157 3073 y Fj(;)17 b(\033)3256 3088 y Fm(2)3295 3073 y Fj(;)g(\033)3394 3088 y Fm(3)3434 3073 y Fi(\))465 b Fy(\(63\))234 3348 y(F)o(or)25 b(the)h(ne)o(xt)e(term)i(corresponding)e(to)h Fj(p)k Fi(=)g(1)c Fy(there)h(are)g(three)g(contrib)n(utions)d(because)j (one)g(can)g(\002nd)f(three)88 3469 y(dif)n(ferent)f(partitions)f(that) i(satisfy)f(the)g(conditions)f(\(24\))1511 3689 y Fj(S)33 b Fi(=)28 b(\(1\))p 1932 3609 66 4 v 99 w Fj(S)34 b Fi(=)27 b(\(2\))99 b Fj(T)42 b Fi(=)27 b(\(3\))1511 3909 y Fj(S)33 b Fi(=)28 b(\(1\))p 1932 3829 V 99 w Fj(S)34 b Fi(=)27 b(\(3\))99 b Fj(T)42 b Fi(=)27 b(\(2\))1511 4083 y Fj(S)33 b Fi(=)28 b(\(2\))p 1932 4003 V 99 w Fj(S)34 b Fi(=)27 b(\(3\))99 b Fj(T)42 b Fi(=)27 b(\(1\))88 4257 y Fy(and)d(according)h (to)f(the)h(formulae)g(\(25,26\))917 4477 y Fj(I)960 4492 y Fm(1)1028 4477 y Fi(=)i Fj(I)1174 4493 y Fm(\(1\))p Fl(;)p Fm(\(2\))p Fl(;)p Fm(\(3\))1510 4477 y Fo(\000)c Fj(I)1653 4493 y Fm(\(1\))p Fl(;)p Fm(\(3\))p Fl(;)p Fm(\(2\))1989 4477 y Fi(+)f Fj(I)2130 4493 y Fm(\(2\))p Fl(;)p Fm(\(3\))p Fl(;)p Fm(\(1\))3937 4477 y Fy(\(64\))88 4697 y(where)473 4943 y Fj(I)516 4959 y Fm(\(1\))p Fl(;)p Fm(\(2\))p Fl(;)p Fm(\(3\))858 4943 y Fi(=)962 4808 y Fn(Z)1017 5033 y Fm(\000)1061 5042 y Fg(1)1126 4876 y Fj(d\033)1232 4891 y Fm(1)p 1126 4920 146 4 v 1128 5012 a Fi(2)p Fj(\031)t(i)1298 4808 y Fn(Z)1354 5033 y Fm(\000)1398 5042 y Fg(3)1463 4876 y Fj(d\033)1569 4891 y Fm(3)p 1463 4920 V 1465 5012 a Fi(2)p Fj(\031)t(i)1619 4943 y(X)1700 4958 y Fm(1)1739 4943 y Fi(\()p Fj(\033)1832 4958 y Fm(1)1872 4943 y Fi(\))p Fj(p)1959 4958 y Fm(2)1998 4943 y Fi(\()p Fj(\033)2091 4958 y Fm(1)2131 4943 y Fi(\)\010\()p Fj(\033)2332 4958 y Fm(1)2372 4943 y Fi(\))p Fj(X)2491 4958 y Fm(3)2531 4943 y Fi(\()p Fj(\033)2624 4958 y Fm(3)2663 4943 y Fi(\)\010\()p Fj(\033)2864 4958 y Fm(3)2905 4943 y Fi(\))p Fj(h)2999 4958 y Fm(1)3038 4943 y Fi(\()p Fj(\033)3131 4958 y Fm(1)3171 4943 y Fo(j)p Fj(\033)3254 4958 y Fm(3)3294 4943 y Fi(\))473 5212 y Fj(I)516 5228 y Fm(\(1\))p Fl(;)p Fm(\(3\))p Fl(;)p Fm(\(2\))858 5212 y Fi(=)962 5077 y Fn(Z)1017 5302 y Fm(\000)1061 5311 y Fg(1)1126 5145 y Fj(d\033)1232 5160 y Fm(1)p 1126 5189 V 1128 5280 a Fi(2)p Fj(\031)t(i)1298 5077 y Fn(Z)1354 5302 y Fm(\000)1398 5311 y Fg(2)1463 5145 y Fj(d\033)1569 5160 y Fm(2)p 1463 5189 V 1465 5280 a Fi(2)p Fj(\031)t(i)1619 5212 y(X)1700 5227 y Fm(1)1739 5212 y Fi(\()p Fj(\033)1832 5227 y Fm(1)1872 5212 y Fi(\))p Fj(p)1959 5227 y Fm(3)1998 5212 y Fi(\()p Fj(\033)2091 5227 y Fm(1)2131 5212 y Fi(\)\010\()p Fj(\033)2332 5227 y Fm(1)2372 5212 y Fi(\))p Fj(X)2491 5227 y Fm(2)2531 5212 y Fi(\()p Fj(\033)2624 5227 y Fm(2)2663 5212 y Fi(\)\010\()p Fj(\033)2864 5227 y Fm(2)2905 5212 y Fi(\))p Fj(h)2999 5227 y Fm(1)3038 5212 y Fi(\()p Fj(\033)3131 5227 y Fm(1)3171 5212 y Fo(j)p Fj(\033)3254 5227 y Fm(2)3294 5212 y Fi(\))473 5481 y Fj(I)516 5497 y Fm(\(2\))p Fl(;)p Fm(\(3\))p Fl(;)p Fm(\(1\))858 5481 y Fi(=)962 5345 y Fn(Z)1017 5571 y Fm(\000)1061 5580 y Fg(1)1126 5414 y Fj(d\033)1232 5429 y Fm(1)p 1126 5458 V 1128 5549 a Fi(2)p Fj(\031)t(i)1298 5345 y Fn(Z)1354 5571 y Fm(\000)1398 5580 y Fg(2)1463 5414 y Fj(d\033)1569 5429 y Fm(2)p 1463 5458 V 1465 5549 a Fi(2)p Fj(\031)t(i)1619 5481 y(X)1700 5496 y Fm(2)1739 5481 y Fi(\()p Fj(\033)1832 5496 y Fm(2)1872 5481 y Fi(\))p Fj(p)1959 5496 y Fm(3)1998 5481 y Fi(\()p Fj(\033)2091 5496 y Fm(2)2131 5481 y Fi(\)\010\()p Fj(\033)2332 5496 y Fm(2)2372 5481 y Fi(\))p Fj(X)2491 5496 y Fm(1)2531 5481 y Fi(\()p Fj(\033)2624 5496 y Fm(1)2663 5481 y Fi(\)\010\()p Fj(\033)2864 5496 y Fm(1)2905 5481 y Fi(\))p Fj(h)2999 5496 y Fm(1)3038 5481 y Fi(\()p Fj(\033)3131 5496 y Fm(2)3171 5481 y Fo(j)p Fj(\033)3254 5496 y Fm(1)3294 5481 y Fi(\))605 b Fy(\(65\))2046 5871 y(19)p eop %%Page: 20 20 20 19 bop 88 289 a Fy(and)24 b(due)h(to)g(the)f(formula)g(\(28\))730 553 y Fj(h)786 568 y Fm(1)826 553 y Fi(\()p Fj(\033)919 568 y Fm(1)958 553 y Fo(j)p Fj(\033)1041 568 y Fm(3)1081 553 y Fi(\))k(=)f Fj(\016)1297 512 y Fk(\000)p Fm(1)1293 578 y Fl(\033)1333 587 y Fg(1)1391 413 y Fn(\022)1465 553 y Fj(R)1539 568 y Fm(1)1578 553 y Fj(h)1634 568 y Fm(0)1674 553 y Fi(\()p Fj(\033)1767 568 y Fm(1)1807 553 y Fj(;)17 b(\033)1906 568 y Fm(2)1945 553 y Fj(;)g(\033)2044 568 y Fm(3)2084 553 y Fi(\))22 b Fo(\000)h Fj(D)2325 568 y Fm(3)2364 553 y Fj(f)2412 568 y Fm(1)2452 553 y Fi(\()p Fj(\033)2545 568 y Fm(1)2585 553 y Fo(j)p Fj(\033)2668 568 y Fm(3)2707 553 y Fi(\))2745 413 y Fn(\023)3937 553 y Fy(\(66\))88 818 y(with)h(the)g(\223e)o(xact)h(form\224)g(gi)n(v)o (en)e(by)i(\(27\))793 1085 y Fj(D)874 1100 y Fm(3)913 1085 y Fj(f)961 1100 y Fm(1)1001 1085 y Fi(\()p Fj(\033)1094 1100 y Fm(1)1134 1085 y Fo(j)p Fj(\033)1217 1100 y Fm(3)1256 1085 y Fi(\))j(=)1462 1017 y Fj(f)1510 1032 y Fm(1)1550 1017 y Fi(\()p Fj(\033)1643 1032 y Fm(1)1682 1017 y Fo(j)p Fj(\033)1765 1032 y Fm(3)1805 1017 y Fi(\))p 1408 1062 490 4 v 1408 1157 a Fj(P)14 b Fi(\()p Fj(\033)1578 1172 y Fm(3)1639 1157 y Fi(+)1747 1118 y Fm(3)p Fl(\031)r(i)p 1747 1134 103 4 v 1781 1191 a Fm(2)1859 1157 y Fi(\))1929 1085 y Fo(\000)2039 1017 y Fj(f)2087 1032 y Fm(1)2126 1017 y Fi(\()p Fj(\033)2219 1032 y Fm(1)2259 1017 y Fo(j)p Fj(\033)2342 1032 y Fm(3)2404 1017 y Fo(\000)23 b Fi(2)p Fj(\031)t(i)p Fi(\))p 2039 1062 644 4 v 2115 1157 a Fj(P)14 b Fi(\()p Fj(\033)2285 1172 y Fm(3)2347 1157 y Fo(\000)2456 1118 y Fm(3)p Fl(\031)r(i)p 2456 1134 103 4 v 2490 1191 a Fm(2)2568 1157 y Fi(\))3937 1085 y Fy(\(67\))234 1363 y(In)23 b(order)g(to)g(de\002ne)g(the)g(function)f Fj(f)1488 1378 y Fm(1)1551 1363 y Fy(we)h(need)g(some)g(f)o(acts)g(about)f(the)h (structure)g(of)g(non-zero)g(residues.)30 b(Let)88 1483 y(us)24 b(introduce)g(the)h(notation)1322 1604 y(Res)1472 1619 y Fl(\033)1512 1628 y Fg(2)1547 1619 y Fm(=)p Fl(\033)1642 1628 y Fg(1)1677 1619 y Fm(+)p Fl(\031)r(i)1803 1604 y Fj(h)1859 1619 y Fm(0)1898 1604 y Fi(\()p Fj(\033)1991 1619 y Fm(1)2031 1604 y Fj(;)17 b(\033)2130 1619 y Fm(2)2170 1604 y Fj(;)g(\033)2269 1619 y Fm(3)2308 1604 y Fi(\))28 b Fo(\021)g Fj(V)2536 1619 y Fm(1)2575 1604 y 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5099 y Fm(1)1314 5084 y Fi(\()p Fj(\033)1407 5099 y Fm(1)1447 5084 y Fi(\)\010\()p Fj(\033)1648 5099 y Fm(1)1688 5084 y Fi(\))p Fj(B)1800 5099 y Fm(23)1875 5084 y Fi(\()p Fj(\033)1968 5099 y Fm(2)2008 5084 y Fi(\))p Fj(\016)2089 5099 y Fl(\033)2129 5108 y Fg(2)2168 5084 y Fj(h)2224 5099 y Fm(1)2264 5084 y Fi(\()p Fj(\033)2357 5099 y Fm(2)2397 5084 y Fo(j)p Fj(\033)2480 5099 y Fm(1)2519 5084 y Fi(\))340 5353 y(=)28 b Fo(\000)538 5217 y Fn(Z)593 5443 y Fm(\000)637 5452 y Fg(1)702 5285 y Fj(d\033)808 5300 y Fm(1)p 702 5330 V 704 5421 a Fi(2)p Fj(\031)t(i)874 5217 y Fn(Z)930 5443 y Fm(\000)974 5452 y Fg(2)1039 5285 y Fj(d\033)1145 5300 y Fm(2)p 1039 5330 V 1041 5421 a Fi(2)p Fj(\031)t(i)1195 5353 y(X)1276 5368 y Fm(1)1315 5353 y Fi(\()p Fj(\033)1408 5368 y Fm(1)1448 5353 y Fi(\)\010\()p Fj(\033)1649 5368 y Fm(1)1689 5353 y Fi(\)\()p Fj(X)1846 5368 y Fm(2)1885 5353 y Fi(\()p Fj(\033)1978 5368 y Fm(2)2018 5353 y Fi(\))p Fj(p)2105 5368 y Fm(3)2145 5353 y Fi(\()p Fj(\033)2238 5368 y Fm(2)2277 5353 y Fi(\))22 b(+)g Fj(X)2516 5368 y Fm(3)2556 5353 y Fi(\()p Fj(\033)2649 5368 y Fm(2)2689 5353 y Fi(\))p Fj(p)2776 5368 y Fm(2)2815 5353 y Fi(\()p Fj(\033)2908 5368 y Fm(2)2948 5353 y Fi(\)\)\010\()p Fj(\033)3187 5368 y Fm(2)3227 5353 y Fi(\))p Fj(h)3321 5368 y Fm(1)3361 5353 y Fi(\()p Fj(\033)3454 5368 y Fm(2)3493 5353 y Fo(j)p Fj(\033)3576 5368 y Fm(1)3616 5353 y Fi(\))340 5569 y(=)28 b Fo(\000)p Fj(I)564 5585 y Fm(\(2\))p Fl(;)p Fm(\(3\))p Fl(;)p Fm(\(1\))900 5569 y Fo(\000)23 b Fj(I)1043 5585 y Fm(\(2\))p Fl(;)p Fm(\(3\))p Fl(;)p Fm(\(1\))1357 5606 y Fk(j)1377 5615 y Fg(2)p Fe($)p Fg(3)3937 5569 y Fy(\(78\))2046 5871 y(21)p eop %%Page: 22 22 22 21 bop 644 326 a Fo(F)716 341 y Fm(1)783 326 y Fi(=)887 190 y Fn(Z)942 416 y Fm(\000)986 425 y Fg(1)1051 258 y Fj(d\033)1157 273 y Fm(1)p 1051 303 146 4 v 1053 394 a Fi(2)p Fj(\031)t(i)1223 190 y Fn(Z)1279 416 y Fm(\000)1323 425 y Fg(2)1388 258 y Fj(d\033)1494 273 y Fm(2)p 1388 303 V 1390 394 a Fi(2)p Fj(\031)t(i)1544 326 y(X)1625 341 y Fm(1)1664 326 y Fi(\()p Fj(\033)1757 341 y Fm(1)1797 326 y Fi(\)\010\()p Fj(\033)1998 341 y Fm(1)2038 326 y Fi(\))p Fj(B)2150 341 y Fm(23)2225 326 y Fi(\()p Fj(\033)2318 341 y Fm(2)2358 326 y Fi(\))p Fj(D)2477 341 y Fm(1)2516 326 y Fj(f)2564 341 y Fm(1)2604 326 y Fi(\()p Fj(\033)2697 341 y Fm(2)2736 326 y Fo(j)p Fj(\033)2819 341 y Fm(1)2859 326 y Fi(\))672 600 y(=)28 b Fo(\000)870 465 y Fn(Z)925 690 y Fm(\000)969 699 y Fg(1)1034 533 y Fj(d\033)1140 548 y Fm(1)p 1034 577 V 1036 668 a Fi(2)p Fj(\031)t(i)1206 465 y Fn(Z)1262 690 y Fm(\000)1306 699 y Fg(2)1371 533 y Fj(d\033)1477 548 y Fm(2)p 1371 577 V 1373 668 a Fi(2)p Fj(\031)t(i)1527 600 y(B)1601 615 y Fm(23)1676 600 y Fi(\()p Fj(\033)1769 615 y Fm(2)1808 600 y Fi(\))2008 533 y Fj(p)2057 548 y Fm(1)2097 533 y Fi(\()p Fj(\033)2190 548 y Fm(1)2229 533 y Fi(\))p Fj(f)2315 548 y Fm(1)2355 533 y Fi(\()p Fj(\033)2448 548 y Fm(2)2488 533 y Fo(j)p Fj(\033)2571 548 y Fm(1)2610 533 y Fi(\))p 1856 577 944 4 v 1856 672 a Fj(P)14 b Fi(\()p Fj(\033)2026 687 y Fm(1)2088 672 y Fi(+)2196 633 y Fl(\031)r(i)p 2196 649 67 4 v 2211 707 a Fm(2)2273 672 y Fi(\))p Fj(P)g Fi(\()p Fj(\033)2481 687 y Fm(1)2542 672 y Fi(+)2650 633 y Fm(3)p Fl(\031)r(i)p 2650 649 103 4 v 2683 707 a Fm(2)2762 672 y Fi(\))2838 600 y(=)27 b(0)947 b Fy(\(79\))88 880 y(Here)25 b(we)g(ha)n(v)o(e)g (got)f(zero)h(because)g(the)g(ratio)1775 1081 y Fj(p)1824 1096 y Fm(1)1864 1081 y Fi(\()p Fj(\033)1957 1096 y Fm(1)1997 1081 y Fi(\))p Fj(f)2083 1096 y Fm(1)2122 1081 y Fi(\()p Fj(\033)2215 1096 y Fm(2)2255 1081 y Fo(j)p Fj(\033)2338 1096 y Fm(1)2377 1081 y Fi(\))p 1624 1126 944 4 v 1624 1221 a Fj(P)14 b Fi(\()p Fj(\033)1794 1236 y Fm(1)1855 1221 y Fi(+)1963 1182 y Fl(\031)r(i)p 1963 1198 67 4 v 1979 1255 a Fm(2)2040 1221 y Fi(\))p Fj(P)g Fi(\()p Fj(\033)2248 1236 y Fm(1)2309 1221 y Fi(+)2417 1182 y Fm(3)p Fl(\031)r(i)p 2417 1198 103 4 v 2451 1255 a Fm(2)2529 1221 y Fi(\))88 1429 y Fy(can)31 b(ha)n(v)o(e)g(poles)g(only)f(for)i (the)f(v)n(ariable)g Fj(\033)1629 1444 y Fm(12)1735 1429 y Fy(and)g(since)g Fj(\033)2201 1444 y Fm(2)2273 1429 y Fy(belongs)f(to)h(the)g(contour)g Fi(\000)3273 1444 y Fm(2)3343 1429 y Fy(which)g(contains)g(the)88 1549 y(contour)24 b Fi(\000)478 1564 y Fm(1)542 1549 y Fy(those)g(poles)h (can)g(be)g(only)f(outside)f(of)i(the)g(contour)f Fi(\000)2454 1564 y Fm(1)2494 1549 y Fy(.)30 b(Thus)25 b(we)g(come)f(to)h(the)f (follo)n(wing)f(result)997 1767 y Fj(J)1051 1782 y Fm(0)1118 1767 y Fi(=)28 b Fo(\000)p Fj(I)1342 1782 y Fm(\(2\))p Fl(;)p Fm(\(3\))p Fl(;)p Fm(\(1\))1678 1767 y Fo(\000)23 b Fj(I)1821 1782 y Fm(\(2\))p Fl(;)p Fm(\(3\))p Fl(;)p Fm(\(1\))2135 1803 y Fk(j)2155 1812 y Fg(2)p Fe($)p Fg(3)3937 1767 y Fy(\(80\))234 1993 y(It)f(is)f(left)g(to)h(note)f(that)g(the)h (combination)e(of)i(tw)o(o)f(residual)g(terms)g(in)h(the)f(r)-5 b(.h.s.)29 b(of)22 b(\(64\),)g(namely)-6 b(,)21 b Fj(I)3701 2008 y Fm(\(1\))p Fl(;)p Fm(\(2\))p Fl(;)p Fm(\(3\))4026 1993 y Fo(\000)88 2113 y Fj(I)131 2129 y Fm(\(1\))p Fl(;)p Fm(\(3\))p Fl(;)p Fm(\(2\))469 2113 y Fy(is)k(already)g(sk)o(e)n (w-symmetric)e(w)-6 b(.r)h(.t.)29 b(the)c(transposition)e Fi(2)k Fo($)g Fi(3)p Fy(.)234 2233 y(Thus)d(we)i(ha)n(v)o(e)e(check)o (ed)i(the)f(sk)o(e)n(w-symmetry)e(for)i(the)g(transposition)e Fi(2)28 b Fo($)f Fi(3)p Fy(.)32 b(F)o(or)25 b(another)g(transposition) 88 2354 y Fi(1)i Fo($)g Fi(2)d Fy(it)g(is)g(a)h(bit)e(more)h(dif)n (\002cult)g(b)n(ut)f(also)h(possible)f(to)h(do.)30 b(The)24 b(strate)o(gy)g(is)g(more)g(or)g(less)g(repeats)g(that)g(for)h(the)88 2474 y(transposition)d Fi(2)28 b Fo($)f Fi(3)p Fy(.)k(W)-8 b(e)25 b(shall)f(not)g(do)h(it)f(here.)234 2595 y(4.)30 b Fa(The)c(case)f Fj(n)j Fi(=)g(5)234 2715 y Fy(It)42 b(is)f(straightforw)o(ard)g(to)h(write)g(do)n(wn)e(the)i(function)f Fj(D)k Fy(from)d(\(23\))f(in)h(the)g(form)f(lik)o(e)h(\(48\))g(for)g Fj(n)59 b Fi(=)88 2835 y(3)40 b Fy(and)g(\(60\))h(for)f Fj(n)57 b Fi(=)f(4)40 b Fy(de\002ning)g(a)h(polynomial)d Fj(g)t Fi(\()p Fj(\033)2150 2850 y Fm(1)2190 2835 y Fj(;)17 b(\033)2289 2850 y Fm(2)2328 2835 y Fj(;)g(\033)2427 2850 y Fm(3)2467 2835 y Fj(;)g(\033)2566 2850 y Fm(4)2605 2835 y Fi(\))41 b Fy(which)f(also)g(depends)f(on)i(the)f(rapidi-)88 2956 y(ties)e Fj(\014)320 2971 y Fm(1)360 2956 y Fj(;)17 b(:)g(:)g(:)e(;)i(\014)633 2971 y Fm(10)708 2956 y Fy(.)73 b(Again)38 b(the)h(main)f(property)h(of)g(this)f(polynomial)f(is)i (that)f(all)h(polynomails)e(of)i(the)g(form)88 3076 y Fj(g)t Fi(\()p Fj(:)17 b(:)g(:)e(;)i(\033)406 3091 y Fm(1)445 3076 y Fj(;)g(:)g(:)g(:)f(;)h(\033)719 3091 y Fm(1)781 3076 y Fi(+)22 b Fj(\031)t(i;)17 b(:)g(:)g(:)o Fi(\))25 b Fy(contain)f(as)h(a)g(di)n(visor)88 3197 y Fj(P)14 b Fi(\()p Fj(\033)258 3212 y Fm(1)298 3197 y Fi(+)385 3157 y Fl(\031)r(i)p 385 3174 67 4 v 400 3231 a Fm(2)461 3197 y Fi(\))p Fy(,)20 b(the)f(polynomials)e Fj(g)t Fi(\()p Fj(:)g(:)g(:)e(;)i(\033)1515 3212 y Fm(1)1555 3197 y Fj(;)g(:)g(:)g(:)e(;)i(\033)1828 3212 y Fm(1)1869 3197 y Fi(+)q Fj(\031)t(i;)g(:)g(:)g(:)e(;)i(\033)2311 3212 y Fm(1)2352 3197 y Fi(+)q(2)p Fj(\031)t(i)p Fi(\))h Fy(are)i(di)n(visible)d(by)h Fj(P)c Fi(\()p Fj(\033)3415 3212 y Fm(1)3455 3197 y Fi(+)3542 3157 y Fl(\031)r(i)p 3542 3174 V 3558 3231 a Fm(2)3619 3197 y Fi(\))p Fj(P)g Fi(\()p Fj(\033)3827 3212 y Fm(1)3867 3197 y Fi(+)3954 3157 y Fm(3)p Fl(\031)r(i)p 3954 3174 103 4 v 3987 3231 a Fm(2)4065 3197 y Fi(\))88 3317 y Fy(etc.)234 3437 y(No)n(w)24 b(the)h(formula)f(\(26\))h(will)f(contain)g(three)h(terms)f(for)h Fj(p)j Fi(=)g(0)p Fj(;)17 b(p)27 b Fi(=)g(1)e Fy(and)g Fj(p)i Fi(=)h(2)p Fy(:)1010 3655 y Fo(L)p Fi(\()p Fj(p)1166 3670 y Fm(1)1205 3655 y Fj(;)17 b(p)1298 3670 y Fm(2)1337 3655 y Fj(;)g(p)1430 3670 y Fm(3)1469 3655 y Fj(;)g(p)1562 3670 y Fm(4)1601 3655 y Fi(\))28 b(=)g Fj(I)1814 3670 y Fm(0)1875 3655 y Fi(+)22 b Fj(I)2016 3670 y Fm(1)2078 3655 y Fi(+)g Fj(I)2219 3670 y Fm(2)3937 3655 y Fy(\(81\))88 3872 y(with)i(polynomials)e Fj(p)856 3887 y Fm(1)896 3872 y Fj(;)17 b(p)989 3887 y Fm(2)1028 3872 y Fj(;)g(p)1121 3887 y Fm(3)1160 3872 y Fj(;)g(p)1253 3887 y Fm(4)1317 3872 y Fy(from)24 b(the)h(set)g Fj(s)1864 3887 y Fk(\000)p Fm(4)1958 3872 y Fj(;)17 b(s)2048 3887 y Fk(\000)p Fm(3)2142 3872 y Fj(;)g(s)2232 3887 y Fk(\000)p Fm(2)2326 3872 y Fj(;)g(s)2416 3887 y Fk(\000)p Fm(1)2510 3872 y Fj(;)g(s)2600 3887 y Fm(1)2639 3872 y Fj(;)g(s)2729 3887 y Fm(2)2768 3872 y Fj(;)g(s)2858 3887 y Fm(3)2897 3872 y Fj(;)g(s)2987 3887 y Fm(4)3026 3872 y Fy(.)234 3992 y(The)25 b(main)f(term)g Fj(I)888 4007 y Fm(0)953 3992 y Fy(corresponds)g(to)g(the)h(partitions) 1522 4210 y Fj(S)33 b Fi(=)p 1719 4130 66 4 v 28 w Fj(S)g Fi(=)28 b Fo(;)99 b Fj(T)42 b Fi(=)27 b(\(1)p Fj(;)17 b Fi(2)p Fj(;)g Fi(3)p Fj(;)g Fi(4\))622 4524 y Fj(I)665 4539 y Fm(0)732 4524 y Fo(\021)28 b Fj(I)880 4540 y Fk(;)p Fl(;)p Fk(;)p Fl(;)p Fm(\(1)p Fl(;)p Fm(2)p Fl(;)p Fm(3)p Fl(;)p Fm(4\))1277 4524 y Fi(=)639 4594 y Fn(Z)694 4820 y Fm(\000)738 4829 y Fg(1)803 4663 y Fj(d\033)909 4678 y Fm(1)p 803 4707 146 4 v 805 4798 a Fi(2)p Fj(\031)t(i)976 4594 y Fn(Z)1031 4820 y Fm(\000)1075 4829 y Fg(2)1140 4663 y Fj(d\033)1246 4678 y Fm(2)p 1140 4707 V 1142 4798 a Fi(2)p Fj(\031)t(i)1312 4594 y Fn(Z)1368 4820 y Fm(\000)1412 4829 y Fg(3)1477 4663 y Fj(d\033)1583 4678 y Fm(3)p 1477 4707 V 1479 4798 a Fi(2)p Fj(\031)t(i)1649 4594 y Fn(Z)1705 4820 y Fm(\000)1749 4829 y Fg(4)1814 4663 y Fj(d\033)1920 4678 y Fm(4)p 1814 4707 V 1816 4798 a Fi(2)p Fj(\031)t(i)1970 4730 y(X)2051 4745 y Fm(1)2090 4730 y Fi(\()p Fj(\033)2183 4745 y Fm(1)2223 4730 y Fi(\)\010\()p Fj(\033)2424 4745 y Fm(1)2464 4730 y Fi(\))p Fj(X)2583 4745 y Fm(2)2622 4730 y Fi(\()p Fj(\033)2715 4745 y Fm(2)2755 4730 y Fi(\)\010\()p Fj(\033)2956 4745 y Fm(2)2996 4730 y Fi(\))644 4947 y Fo(\001)22 b Fj(X)775 4962 y Fm(3)814 4947 y Fi(\()p Fj(\033)907 4962 y Fm(3)947 4947 y Fi(\)\010\()p Fj(\033)1148 4962 y Fm(3)1188 4947 y Fi(\))p Fj(X)1307 4962 y Fm(4)1347 4947 y Fi(\()p Fj(\033)1440 4962 y Fm(4)1480 4947 y Fi(\)\010\()p Fj(\033)1681 4962 y Fm(4)1721 4947 y Fi(\))p Fj(h)1815 4962 y Fm(0)1854 4947 y Fi(\()p Fj(\033)1947 4962 y Fm(1)1987 4947 y Fj(;)17 b(\033)2086 4962 y Fm(2)2126 4947 y Fj(;)g(\033)2225 4962 y Fm(3)2264 4947 y Fj(;)g(\033)2363 4962 y Fm(4)2403 4947 y Fi(\))1496 b Fy(\(82\))88 5164 y(where)25 b(as)g(abo)o(v)o(e)f (the)g(function)g Fj(h)1283 5179 y Fm(0)1323 5164 y Fi(\()p Fj(\033)1416 5179 y Fm(1)1456 5164 y Fj(;)17 b(:)g(:)g(:)e(;)i(\033) 1729 5179 y Fm(4)1769 5164 y Fi(\))25 b Fy(is)f(de\002ned)h(by)g(the)g (formula)f(\(22\).)234 5284 y(F)o(or)k(the)g(ne)o(xt)g(term)g(with)f Fj(p)34 b Fi(=)g(1)29 b Fy(there)f(are)h(six)f(contrib)n(utions)e (corresponding)h(to)h(six)g(dif)n(ferent)g(partitions)88 5405 y(that)c(satisfy)g(the)h(conditions)e(\(24\))1486 5622 y Fj(S)34 b Fi(=)27 b(\(1\))p 1908 5542 66 4 v 100 w Fj(S)33 b Fi(=)28 b(\(2\))99 b Fj(T)41 b Fi(=)28 b(\(34\))2046 5871 y Fy(22)p eop %%Page: 23 23 23 22 bop 1486 289 a Fj(S)34 b Fi(=)27 b(\(1\))p 1908 209 66 4 v 100 w Fj(S)33 b Fi(=)28 b(\(3\))99 b Fj(T)41 b Fi(=)28 b(\(24\))1486 455 y Fj(S)34 b Fi(=)27 b(\(1\))p 1908 375 V 100 w Fj(S)33 b Fi(=)28 b(\(4\))99 b Fj(T)41 b Fi(=)28 b(\(23\))1486 621 y Fj(S)34 b Fi(=)27 b(\(2\))p 1908 541 V 100 w Fj(S)33 b Fi(=)28 b(\(3\))99 b Fj(T)41 b Fi(=)28 b(\(14\))1486 787 y Fj(S)34 b Fi(=)27 b(\(2\))p 1908 707 V 100 w Fj(S)33 b Fi(=)28 b(\(4\))99 b Fj(T)41 b Fi(=)28 b(\(13\))1486 954 y Fj(S)34 b Fi(=)27 b(\(3\))p 1908 874 V 100 w Fj(S)33 b Fi(=)28 b(\(4\))99 b Fj(T)41 b Fi(=)28 b(\(12\))88 1120 y Fy(and)c(according)h(to)f(the)h(formulae)g (\(25,26\))314 1321 y Fj(I)357 1336 y Fm(1)424 1321 y Fi(=)j Fj(I)571 1336 y Fm(\(1\))p Fl(;)p Fm(\(2\))p Fl(;)p Fm(\(34\))942 1321 y Fo(\000)23 b Fj(I)1085 1336 y Fm(\(1\))p Fl(;)p Fm(\(3\))p Fl(;)p Fm(\(24\))1456 1321 y Fi(+)f Fj(I)1597 1336 y Fm(\(1\))p Fl(;)p Fm(\(4\))p Fl(;)p Fm(\(23\))1968 1321 y Fi(+)g Fj(I)2109 1336 y Fm(\(2\))p Fl(;)p Fm(\(3\))p Fl(;)p Fm(\(14\))2481 1321 y Fo(\000)g Fj(I)2623 1336 y Fm(\(2\))p Fl(;)p Fm(\(4\))p Fl(;)p Fm(\(13\))2994 1321 y Fi(+)g Fj(I)3135 1336 y Fm(\(3\))p Fl(;)p Fm(\(4\))p Fl(;)p Fm(\(12\))3937 1321 y Fy(\(83\))88 1522 y(where)533 1749 y Fj(I)576 1764 y Fm(\()p Fl(i)627 1773 y Fg(1)662 1764 y Fm(\))p Fl(;)p Fm(\()p Fl(i)760 1773 y Fg(2)795 1764 y Fm(\))p Fl(;)p Fm(\()p Fl(i)893 1773 y Fg(3)940 1764 y Fl(i)964 1773 y Fg(4)999 1764 y Fm(\))1058 1749 y Fi(=)1161 1613 y Fn(Z)1217 1839 y Fm(\000)1261 1849 y Ff(i)1283 1864 y Fg(1)1352 1682 y Fj(d\033)1458 1697 y Fl(i)1482 1706 y Fg(1)p 1352 1726 169 4 v 1366 1817 a Fi(2)p Fj(\031)t(i)1548 1613 y Fn(Z)1603 1839 y Fm(\000)1647 1849 y Ff(i)1669 1864 y Fg(3)1739 1682 y Fj(d\033)1845 1697 y Fl(i)1869 1706 y Fg(3)p 1739 1726 V 1753 1817 a Fi(2)p Fj(\031)t(i)1935 1613 y Fn(Z)1990 1839 y Fm(\000)2034 1849 y Ff(i)2056 1864 y Fg(4)2126 1682 y Fj(d\033)2232 1697 y Fl(i)2256 1706 y Fg(4)p 2126 1726 V 2140 1817 a Fi(2)p Fj(\031)t(i)2304 1749 y(X)2385 1764 y Fl(i)2409 1773 y Fg(1)2448 1749 y Fi(\()p Fj(\033)2541 1764 y Fl(i)2565 1773 y Fg(1)2604 1749 y Fi(\))p Fj(p)2691 1764 y Fl(i)2715 1773 y Fg(2)2754 1749 y Fi(\()p Fj(\033)2847 1764 y Fl(i)2871 1773 y Fg(1)2910 1749 y Fi(\)\010\()p Fj(\033)3111 1764 y Fl(i)3135 1773 y Fg(1)3175 1749 y Fi(\))555 1981 y Fo(\001)g Fj(X)686 1996 y Fl(i)710 2005 y Fg(3)749 1981 y Fi(\()p Fj(\033)842 1996 y Fl(i)866 2005 y Fg(3)905 1981 y Fi(\)\010\()p Fj(\033)1106 1996 y Fl(i)1130 2005 y Fg(3)1169 1981 y Fi(\))p Fj(X)1288 1996 y Fl(i)1312 2005 y Fg(4)1351 1981 y Fi(\()p Fj(\033)1444 1996 y Fl(i)1468 2005 y Fg(4)1507 1981 y Fi(\)\010\()p Fj(\033)1708 1996 y Fl(i)1732 2005 y Fg(4)1772 1981 y Fi(\))p Fj(h)1866 1996 y Fm(1)1905 1981 y Fi(\()p Fj(\033)1998 1996 y Fl(i)2022 2005 y Fg(1)2061 1981 y Fo(j)p Fj(\033)2144 1996 y Fl(i)2168 2005 y Fg(3)2207 1981 y Fj(;)17 b(\033)2306 1996 y Fl(i)2330 2005 y Fg(4)2369 1981 y Fi(\))1530 b Fy(\(84\))88 2183 y(and)24 b(due)h(to)g(the)f(formula)g(\(28\))294 2434 y Fj(h)350 2449 y Fm(1)389 2434 y Fi(\()p Fj(\033)482 2449 y Fm(1)522 2434 y Fo(j)p Fj(\033)605 2449 y Fm(3)644 2434 y Fj(;)17 b(\033)743 2449 y Fm(4)783 2434 y Fi(\))28 b(=)f Fj(\016)999 2393 y Fk(\000)p Fm(1)995 2458 y Fl(\033)1035 2467 y Fg(1)1093 2293 y Fn(\022)1167 2434 y Fj(R)1241 2449 y Fm(1)1280 2434 y Fj(h)1336 2449 y Fm(0)1376 2434 y Fi(\()p Fj(\033)1469 2449 y Fm(1)1509 2434 y Fj(;)17 b(\033)1608 2449 y Fm(2)1647 2434 y Fj(;)g(\033)1746 2449 y Fm(3)1786 2434 y Fj(;)g(\033)1885 2449 y Fm(4)1925 2434 y Fi(\))22 b Fo(\000)g Fj(D)2165 2449 y Fm(3)2205 2434 y Fj(f)2253 2449 y Fm(1)2292 2434 y Fi(\()p Fj(\033)2385 2449 y Fm(1)2425 2434 y Fo(j)p Fj(\033)2508 2449 y Fm(3)2548 2434 y Fo(j)p Fj(\033)2631 2449 y Fm(4)2670 2434 y Fi(\))g(+)g Fj(D)2909 2449 y Fm(4)2949 2434 y Fj(f)2997 2449 y Fm(1)3036 2434 y Fi(\()p Fj(\033)3129 2449 y Fm(1)3169 2434 y Fo(j)p Fj(\033)3252 2449 y Fm(4)3292 2434 y Fo(j)p Fj(\033)3375 2449 y Fm(3)3414 2434 y Fi(\))3452 2293 y Fn(\023)3937 2434 y Fy(\(85\))88 2692 y(with)34 b(the)h(\223e)o(xact)h(forms\224)f Fj(D)1139 2707 y Fm(3)1213 2692 y Fy(and)h Fj(D)1474 2707 y Fm(4)1549 2692 y Fy(de\002ned)f(by)g(\(27\).)62 b(Let)36 b(us)f(note)g(that)f(the)i(function)e Fj(h)3530 2707 y Fm(1)3570 2692 y Fi(\()p Fj(\033)3663 2707 y Fm(1)3702 2692 y Fo(j)p Fj(\033)3785 2707 y Fm(3)3825 2692 y Fj(;)17 b(\033)3924 2707 y Fm(4)3964 2692 y Fi(\))35 b Fy(is)88 2812 y(manifestly)23 b(anti-symmetric)g(w)-6 b(.r)h(.t.)30 b(the)24 b(transposition)f Fj(\033)2129 2827 y Fm(3)2197 2812 y Fo($)k Fj(\033)2379 2827 y Fm(4)2419 2812 y Fy(.)234 2933 y(F)o(or)h(the)f(last)g(term)h(with)f Fj(p)33 b Fi(=)g(2)28 b Fy(there)g(are)g(three)g(contrib)n(utions)e(which)h (correspond)h(to)f(the)h(three)g(possible)88 3053 y(partitions)23 b(that)h(satisfy)g(the)h(conditions)e(\(24\))1499 3254 y Fj(S)34 b Fi(=)28 b(\(12\))p 1970 3174 66 4 v 99 w Fj(S)33 b Fi(=)28 b(\(34\))99 b Fj(T)41 b Fi(=)28 b Fo(;)1499 3455 y Fj(S)34 b Fi(=)28 b(\(12\))p 1970 3375 V 99 w Fj(S)33 b Fi(=)28 b(\(43\))99 b Fj(T)41 b Fi(=)28 b Fo(;)1499 3622 y Fj(S)34 b Fi(=)28 b(\(13\))p 1970 3542 V 99 w Fj(S)33 b Fi(=)28 b(\(24\))99 b Fj(T)41 b Fi(=)28 b Fo(;)88 3788 y Fy(and)876 3989 y Fj(I)919 4004 y Fm(2)987 3989 y Fi(=)f Fo(\000)p Fj(I)1210 4004 y Fm(\(12\))p Fl(;)p Fm(\(34\))p Fl(;)p Fk(;)1562 3989 y Fi(+)22 b Fj(I)1703 4004 y Fm(\(12\))p Fl(;)p Fm(\(43\))p Fl(;)p Fk(;)2055 3989 y Fi(+)g Fj(I)2196 4004 y Fm(\(13\))p Fl(;)p Fm(\(24\))p Fl(;)p Fk(;)3937 3989 y Fy(\(86\))88 4190 y(where)664 4417 y Fj(I)707 4432 y Fm(\()p Fl(i)758 4441 y Fg(1)805 4432 y Fl(i)829 4441 y Fg(2)864 4432 y Fm(\))p Fl(;)p Fm(\()p Fl(i)962 4441 y Fg(3)1009 4432 y Fl(i)1033 4441 y Fg(4)1067 4432 y Fm(\))p Fl(;)p Fk(;)1181 4417 y Fi(=)1285 4281 y Fn(Z)1340 4507 y Fm(\000)1384 4517 y Ff(i)1406 4532 y Fg(1)1476 4349 y Fj(d\033)1582 4364 y Fl(i)1606 4373 y Fg(1)p 1476 4394 169 4 v 1490 4485 a Fi(2)p Fj(\031)t(i)1672 4281 y Fn(Z)1727 4507 y Fm(\000)1771 4517 y Ff(i)1793 4532 y Fg(2)1863 4349 y Fj(d\033)1969 4364 y Fl(i)1993 4373 y Fg(2)p 1863 4394 V 1877 4485 a Fi(2)p Fj(\031)t(i)2042 4417 y(X)2123 4432 y Fl(i)2147 4441 y Fg(1)2185 4417 y Fi(\()p Fj(\033)2278 4432 y Fl(i)2302 4441 y Fg(1)2341 4417 y Fi(\))p Fj(p)2428 4432 y Fl(i)2452 4441 y Fg(3)2491 4417 y Fi(\()p Fj(\033)2584 4432 y Fl(i)2608 4441 y Fg(1)2647 4417 y Fi(\)\010\()p Fj(\033)2848 4432 y Fl(i)2872 4441 y Fg(1)2912 4417 y Fi(\))687 4649 y Fo(\001)f Fj(X)817 4664 y Fl(i)841 4673 y Fg(2)880 4649 y Fi(\()p Fj(\033)973 4664 y Fl(i)997 4673 y Fg(2)1036 4649 y Fi(\))p Fj(p)1123 4664 y Fl(i)1147 4673 y Fg(4)1186 4649 y Fi(\()p Fj(\033)1279 4664 y Fl(i)1303 4673 y Fg(2)1342 4649 y Fi(\)\010\()p Fj(\033)1543 4664 y Fl(i)1567 4673 y Fg(2)1607 4649 y Fi(\))p Fj(h)1701 4664 y Fm(2)1740 4649 y Fi(\()p Fj(\033)1833 4664 y Fl(i)1857 4673 y Fg(1)1896 4649 y Fj(;)c(\033)1995 4664 y Fl(i)2019 4673 y Fg(2)2058 4649 y Fi(\))1841 b Fy(\(87\))88 4850 y(and)24 b(due)h(to)g(the)f(formula)g(\(28\))826 5102 y Fj(h)882 5117 y Fm(2)921 5102 y Fi(\()p Fj(\033)1014 5117 y Fm(1)1054 5102 y Fj(;)17 b(\033)1153 5117 y Fm(3)1193 5102 y Fi(\))27 b(=)h Fj(\016)1409 5060 y Fk(\000)p Fm(1)1405 5126 y Fl(\033)1445 5135 y Fg(1)1503 5102 y Fj(\016)1550 5060 y Fk(\000)p Fm(1)1546 5126 y Fl(\033)1586 5135 y Fg(3)1644 4961 y Fn(\022)1718 5102 y Fj(R)1792 5117 y Fm(1)1831 5102 y Fj(R)1905 5117 y Fm(3)1945 5102 y Fj(h)2001 5117 y Fm(0)2041 5102 y Fi(\()p Fj(\033)2134 5117 y Fm(1)2173 5102 y Fj(;)17 b(\033)2272 5117 y Fm(2)2312 5102 y Fj(;)g(\033)2411 5117 y Fm(3)2451 5102 y Fj(;)g(\033)2550 5117 y Fm(4)2589 5102 y Fi(\))848 5314 y Fo(\000)22 b Fj(R)1021 5329 y Fm(1)1061 5314 y Fi([)p Fj(D)1169 5329 y Fm(1)1209 5314 y Fj(f)1257 5329 y Fm(1)1296 5314 y Fi(\()p Fj(\033)1389 5329 y Fm(3)1429 5314 y Fo(j)p Fj(\033)1512 5329 y Fm(1)1552 5314 y Fo(j)p Fj(\033)1635 5329 y Fm(2)1674 5314 y Fi(\))g Fo(\000)h Fj(D)1915 5329 y Fm(2)1954 5314 y Fj(f)2002 5329 y Fm(1)2042 5314 y Fi(\()p Fj(\033)2135 5329 y Fm(3)2175 5314 y Fo(j)p Fj(\033)2258 5329 y Fm(2)2297 5314 y Fo(j)p Fj(\033)2380 5329 y Fm(1)2420 5314 y Fi(\)])848 5527 y Fo(\000)f Fj(R)1021 5542 y Fm(3)1061 5527 y Fi([)p Fj(D)1169 5542 y Fm(3)1209 5527 y Fj(f)1257 5542 y Fm(1)1296 5527 y Fi(\()p Fj(\033)1389 5542 y Fm(1)1429 5527 y Fo(j)p Fj(\033)1512 5542 y Fm(3)1552 5527 y Fo(j)p Fj(\033)1635 5542 y Fm(4)1674 5527 y Fi(\))g Fo(\000)h Fj(D)1915 5542 y Fm(4)1954 5527 y Fj(f)2002 5542 y Fm(1)2042 5527 y Fi(\()p Fj(\033)2135 5542 y Fm(1)2175 5527 y Fo(j)p Fj(\033)2258 5542 y Fm(4)2297 5527 y Fo(j)p Fj(\033)2380 5542 y Fm(3)2420 5527 y Fi(\)])2485 5387 y Fn(\023)3937 5527 y Fy(\(88\))2046 5871 y(23)p eop %%Page: 24 24 24 23 bop 88 289 a Fy(Actually)24 b(this)f(formula)i(for)g Fj(h)1162 304 y Fm(2)1201 289 y Fi(\()p Fj(\033)1294 304 y Fm(1)1334 289 y Fj(;)17 b(\033)1433 304 y Fm(3)1473 289 y Fi(\))25 b Fy(is)f(symmetric)g(w)-6 b(.r)h(.t.)29 b(the)c(transposition)d Fj(\033)3040 304 y Fm(1)3108 289 y Fo($)27 b Fj(\033)3290 304 y Fm(3)3330 289 y Fy(.)234 409 y(In)f(order)h(to)f(de\002ne)h(the)f(function)f Fj(f)1508 424 y Fm(1)1574 409 y Fy(we)i(need)f(notation)f(which)h(are)h(similar)e (to)h(those)g(for)h(the)f(case)h Fj(n)j Fi(=)h(4)p Fy(.)88 529 y(Let)24 b(us)h(write)f(only)g(non-zero)h(residues)1184 742 y(Res)1333 757 y Fl(\033)1373 766 y Fg(2)1408 757 y Fm(=)p Fl(\033)1503 766 y Fg(1)1538 757 y Fm(+)p Fl(\031)r(i)1664 742 y Fj(h)1720 757 y Fm(0)1760 742 y Fi(\()p Fj(\033)1853 757 y Fm(1)1892 742 y Fj(;)17 b(\033)1991 757 y Fm(2)2031 742 y Fj(;)g(\033)2130 757 y Fm(3)2170 742 y Fj(;)g(\033)2269 757 y Fm(4)2308 742 y Fi(\))28 b Fo(\021)g Fj(V)2536 757 y Fm(1)2575 742 y Fi(\()p Fj(\033)2668 757 y Fm(1)2708 742 y Fo(j)p Fj(\033)2791 757 y Fm(3)2831 742 y Fj(;)17 b(\033)2930 757 y Fm(4)2969 742 y Fi(\))1078 955 y Fy(Res)1227 970 y Fl(\033)1267 979 y Fg(2)1302 970 y Fm(=)p Fl(\033)1397 979 y Fg(1)1432 970 y Fk(\000)p Fl(\031)r(i)1558 955 y Fj(h)1614 970 y Fm(0)1653 955 y Fi(\()p Fj(\033)1746 970 y Fm(1)1786 955 y Fj(;)g(\033)1885 970 y Fm(2)1925 955 y Fj(;)g(\033)2024 970 y Fm(3)2063 955 y Fj(;)g(\033)2162 970 y Fm(4)2202 955 y Fi(\))28 b(=)f Fj(V)2428 970 y Fm(1)2467 955 y Fi(\()p Fj(\033)2560 970 y Fm(1)2622 955 y Fo(\000)c Fj(\031)t(i)p Fo(j)p Fj(\033)2897 970 y Fm(3)2937 955 y Fj(;)17 b(\033)3036 970 y Fm(4)3075 955 y Fi(\))819 1219 y Fy(Res)969 1234 y Fl(\033)1009 1243 y Fg(3)1044 1234 y Fm(=)p Fl(\033)1139 1243 y Fg(1)1173 1234 y Fm(+2)p Fl(\031)r(i)1335 1219 y Fj(V)1392 1234 y Fm(1)1431 1219 y Fi(\()p Fj(\033)1524 1234 y Fm(1)1564 1219 y Fo(j)p Fj(\033)1647 1234 y Fm(3)1686 1219 y Fj(;)g(\033)1785 1234 y Fm(4)1825 1219 y Fi(\))28 b Fo(\021)g Fj(V)2053 1234 y Fm(2)2092 1219 y Fi(\()p Fj(\033)2185 1234 y Fm(1)2225 1219 y Fo(j)p Fj(\033)2308 1234 y Fm(4)2347 1219 y Fi(\))819 1364 y Fy(Res)969 1379 y Fl(\033)1009 1388 y Fg(3)1044 1379 y Fm(=)p Fl(\033)1139 1388 y Fg(1)1173 1379 y Fk(\000)p Fl(\031)r(i)1299 1364 y Fj(V)1356 1379 y Fm(1)1396 1364 y Fi(\()p Fj(\033)1489 1379 y Fm(1)1529 1364 y Fo(j)p Fj(\033)1612 1379 y Fm(3)1651 1364 y Fj(;)17 b(\033)1750 1379 y Fm(4)1790 1364 y Fi(\))27 b(=)h Fo(\000)p Fj(V)2093 1379 y Fm(2)2133 1364 y Fi(\()p Fj(\033)2226 1379 y Fm(1)2288 1364 y Fo(\000)22 b Fj(\031)t(i)p Fo(j)p Fj(\033)2562 1379 y Fm(4)2602 1364 y Fi(\))1297 b Fy(\(89\))932 1669 y(Res)1082 1684 y Fl(\033)1122 1693 y Fg(4)1157 1684 y Fm(=)p Fl(\033)1252 1693 y Fg(1)1286 1684 y Fm(+3)p Fl(\031)r(i)1448 1669 y Fj(V)1505 1684 y Fm(2)1544 1669 y Fi(\()p Fj(\033)1637 1684 y Fm(1)1677 1669 y Fo(j)p Fj(\033)1760 1684 y Fm(4)1799 1669 y Fi(\))28 b Fo(\021)g Fj(V)2027 1684 y Fm(3)2066 1669 y Fi(\()p Fj(\033)2159 1684 y Fm(1)2199 1669 y Fi(\))932 1815 y Fy(Res)1082 1830 y Fl(\033)1122 1839 y Fg(4)1157 1830 y Fm(=)p Fl(\033)1252 1839 y Fg(1)1286 1830 y Fm(+)p Fl(\031)r(i)1412 1815 y Fj(V)1469 1830 y Fm(2)1509 1815 y Fi(\()p Fj(\033)1602 1830 y Fm(1)1641 1815 y Fo(j)p Fj(\033)1724 1830 y Fm(4)1764 1815 y Fi(\))g Fo(\021)g Fj(V)1992 1830 y Fm(2)p Fl(;)p Fm(1)2086 1815 y Fi(\()p Fj(\033)2179 1830 y Fm(1)2219 1815 y Fi(\))932 1960 y Fy(Res)1082 1975 y Fl(\033)1122 1984 y Fg(4)1157 1975 y Fm(=)p Fl(\033)1252 1984 y Fg(1)1286 1975 y Fk(\000)p Fl(\031)r(i)1412 1960 y Fj(V)1469 1975 y Fm(2)1509 1960 y Fi(\()p Fj(\033)1602 1975 y Fm(1)1641 1960 y Fo(j)p Fj(\033)1724 1975 y Fm(4)1764 1960 y Fi(\))g(=)f Fj(V)1990 1975 y Fm(3)2029 1960 y Fi(\()p Fj(\033)2122 1975 y Fm(1)2184 1960 y Fo(\000)c Fj(\031)t(i)p Fi(\))1523 b Fy(\(90\))234 2173 y(The)25 b(result)f(for)h(the)g(function)f Fj(f)1353 2188 y Fm(1)1417 2173 y Fy(looks)g(as)h(follo)n(ws:)952 2435 y Fj(f)1000 2450 y Fm(1)1039 2435 y Fi(\()p Fj(\033)1132 2450 y Fm(1)1172 2435 y Fo(j)p Fj(\033)1255 2450 y Fm(3)1295 2435 y Fo(j)p Fj(\033)1378 2450 y Fm(4)1417 2435 y Fi(\))j(=)1596 2368 y Fj(V)1653 2383 y Fm(2)1693 2368 y Fi(\()p Fj(\033)1786 2383 y Fm(3)1825 2368 y Fo(j)p Fj(\033)1908 2383 y Fm(4)1948 2368 y Fi(\))p 1596 2412 390 4 v 1726 2503 a Fj(\033)1781 2518 y Fm(13)2018 2435 y Fi(+)22 b(\()2205 2368 y(1)p 2164 2412 131 4 v 2164 2503 a Fj(\033)2219 2518 y Fm(13)2326 2435 y Fo(\000)2584 2368 y Fi(1)p 2436 2412 345 4 v 2436 2503 a Fj(\033)2491 2518 y Fm(13)2588 2503 y Fo(\000)h Fj(\031)t(i)2790 2435 y Fi(\))2900 2368 y Fj(V)2957 2383 y 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y(which)g(can)h(be)g(v)o(eri\002ed)g(straightforw)o (ardly)e(using)h(the)h(abo)o(v)o(e)f(properties)g(\(92,93\))g(of)h(the) g(function)f Fj(f)3727 4181 y Fm(1)464 4433 y Fj(R)538 4448 y Fm(1)578 4292 y Fn(\022)651 4433 y Fj(D)732 4448 y Fm(1)771 4433 y Fj(f)819 4448 y Fm(1)859 4433 y Fi(\()p Fj(\033)952 4448 y Fm(3)992 4433 y Fo(j)p Fj(\033)1075 4448 y Fm(1)1114 4433 y Fo(j)p Fj(\033)1197 4448 y Fm(2)1237 4433 y Fi(\))e Fo(\000)h Fj(D)1478 4448 y Fm(2)1517 4433 y Fj(f)1565 4448 y Fm(1)1605 4433 y Fi(\()p Fj(\033)1698 4448 y Fm(3)1737 4433 y Fo(j)p Fj(\033)1820 4448 y Fm(2)1860 4433 y Fo(j)p Fj(\033)1943 4448 y Fm(1)1982 4433 y Fi(\))2020 4292 y Fn(\023)2121 4433 y Fi(=)28 b Fj(\016)2268 4448 y Fl(\033)2308 4457 y Fg(1)2347 4292 y Fn(\022)2582 4366 y Fj(\037)2643 4381 y Fm(1)2682 4366 y Fi(\()p Fj(\033)2775 4381 y Fm(3)2815 4366 y Fo(j)p Fj(\033)2898 4381 y Fm(1)2960 4366 y Fo(\000)22 b Fj(\031)t(i)p Fi(\))p 2431 4410 910 4 v 2431 4505 a Fj(P)14 b Fi(\()p Fj(\033)2601 4520 y Fm(1)2662 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y Fm(2)3888 3353 y Fy(\(100\))88 3555 y(where)342 3782 y Fj(J)405 3741 y Fk(0)396 3807 y Fm(0)463 3782 y Fi(=)567 3647 y Fn(Z)622 3872 y Fm(\000)666 3881 y Fg(1)731 3715 y Fj(d\033)837 3730 y Fm(1)p 731 3759 146 4 v 733 3851 a Fi(2)p Fj(\031)t(i)903 3647 y Fn(Z)959 3872 y Fm(\000)1003 3881 y Fg(2)1068 3715 y Fj(d\033)1174 3730 y Fm(2)p 1068 3759 V 1070 3851 a Fi(2)p Fj(\031)t(i)1240 3647 y Fn(Z)1296 3872 y Fm(\000)1340 3881 y Fg(3)1405 3715 y Fj(d\033)1511 3730 y Fm(3)p 1405 3759 V 1407 3851 a Fi(2)p Fj(\031)t(i)1561 3782 y(X)1642 3797 y Fm(1)1681 3782 y Fi(\()p Fj(\033)1774 3797 y Fm(1)1814 3782 y Fi(\)\010\()p Fj(\033)2015 3797 y Fm(1)2055 3782 y Fi(\))p Fj(X)2174 3797 y Fm(2)2213 3782 y Fi(\()p Fj(\033)2306 3797 y Fm(2)2346 3782 y Fi(\)\010\()p Fj(\033)2547 3797 y Fm(2)2587 3782 y Fi(\))p Fj(B)2699 3797 y Fm(34)2774 3782 y Fi(\()p Fj(\033)2867 3797 y Fm(3)2907 3782 y Fi(\))p Fj(\016)2988 3797 y Fl(\033)3028 3806 y Fg(3)3067 3782 y Fj(h)3123 3797 y Fm(1)3163 3782 y Fi(\()p 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b Fy(\(102\))88 5312 y(The)24 b(reason)h(why)f(we)h(get)g(zero)g(here)h(is)e(similar)g (to)g(that)h(of)f(the)h(pre)n(vious)f(case,)h(namely)-6 b(,)23 b(because)j(the)e(ratio)1714 5498 y Fj(p)1763 5513 y Fm(1)1803 5498 y Fi(\()p Fj(\033)1896 5513 y Fm(1)1935 5498 y Fi(\))p Fj(f)2021 5513 y Fm(1)2061 5498 y Fi(\()p Fj(\033)2154 5513 y Fm(3)2194 5498 y Fo(j)p Fj(\033)2277 5513 y Fm(1)2316 5498 y Fo(j)p Fj(\033)2399 5513 y Fm(2)2439 5498 y Fi(\))p 1624 5543 944 4 v 1624 5638 a Fj(P)14 b Fi(\()p Fj(\033)1794 5653 y Fm(1)1855 5638 y Fi(+)1963 5598 y Fl(\031)r(i)p 1963 5615 67 4 v 1979 5672 a Fm(2)2040 5638 y Fi(\))p Fj(P)g Fi(\()p Fj(\033)2248 5653 y Fm(1)2309 5638 y Fi(+)2417 5598 y Fm(3)p Fl(\031)r(i)p 2417 5615 103 4 v 2451 5672 a Fm(2)2529 5638 y Fi(\))2046 5871 y Fy(25)p eop %%Page: 26 26 26 25 bop 88 289 a Fy(can)25 b(ha)n(v)o(e)f(poles)g(only)g(for)h(the)g (v)n(ariables)f Fj(\033)1622 304 y Fm(12)1722 289 y Fy(and)g Fj(\033)1945 304 y Fm(13)2020 289 y Fy(.)31 b(In)25 b(both)f(cases)h (these)f(poles)h(are)g(outside)f(of)g(the)h(contour)88 409 y Fi(\000)149 424 y Fm(1)188 409 y Fy(.)234 529 y(The)g(situation)e (with)h(the)h(third)f(term)g(in)h(\(100\))f Fo(F)1966 544 y Fm(2)2030 529 y Fy(is)h(dif)n(ferent)313 792 y Fo(F)385 807 y Fm(2)452 792 y Fi(=)j Fo(\000)650 656 y Fn(Z)705 882 y Fm(\000)749 891 y Fg(1)814 724 y Fj(d\033)920 739 y Fm(1)p 814 769 146 4 v 816 860 a Fi(2)p Fj(\031)t(i)987 656 y Fn(Z)1042 882 y Fm(\000)1086 891 y Fg(2)1151 724 y Fj(d\033)1257 739 y Fm(2)p 1151 769 V 1153 860 a Fi(2)p Fj(\031)t(i)1323 656 y Fn(Z)1379 882 y Fm(\000)1423 891 y Fg(3)1488 724 y Fj(d\033)1594 739 y Fm(3)p 1488 769 V 1490 860 a Fi(2)p Fj(\031)t(i)1644 792 y(X)1725 807 y Fm(1)1764 792 y Fi(\()p Fj(\033)1857 807 y Fm(1)1897 792 y Fi(\)\010\()p Fj(\033)2098 807 y Fm(1)2138 792 y Fi(\))p Fj(X)2257 807 y Fm(2)2296 792 y Fi(\()p Fj(\033)2389 807 y Fm(2)2429 792 y Fi(\)\010\()p Fj(\033)2630 807 y Fm(2)2670 792 y Fi(\))p Fj(B)2782 807 y Fm(34)2857 792 y Fi(\()p Fj(\033)2950 807 y 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1673 y Fi(\))p Fj(f)1348 1688 y Fm(1)1387 1673 y Fi(\()p Fj(\033)1480 1688 y Fm(3)1520 1673 y Fo(j)p Fj(\033)1603 1688 y Fm(2)1642 1673 y Fo(j)p Fj(\033)1725 1688 y Fm(1)1765 1673 y Fi(\))p 950 1718 944 4 v 950 1813 a Fj(P)14 b Fi(\()p Fj(\033)1120 1828 y Fm(2)1181 1813 y Fi(+)1289 1773 y Fl(\031)r(i)p 1289 1790 67 4 v 1305 1847 a Fm(2)1366 1813 y Fi(\))p Fj(P)g Fi(\()p Fj(\033)1574 1828 y Fm(2)1635 1813 y Fi(+)1743 1773 y Fm(3)p Fl(\031)r(i)p 1743 1790 103 4 v 1777 1847 a Fm(2)1856 1813 y Fi(\))631 2025 y(=)27 b Fj(D)815 2040 y Fm(1)855 1884 y Fn(\022)928 2025 y Fj(p)977 2040 y Fm(2)1016 2025 y Fi(\()p Fj(\033)1109 2040 y Fm(1)1149 2025 y Fi(\))1197 1957 y Fj(\037)1258 1972 y Fm(3)1298 1957 y Fi(\()p Fj(\033)1391 1972 y Fm(3)1430 1957 y Fo(j)p Fj(\033)1513 1972 y Fm(1)1575 1957 y Fo(\000)c Fj(\031)t(i)p Fi(\))p 1197 2002 608 4 v 1273 2097 a Fj(P)14 b Fi(\()p Fj(\033)1443 2112 y Fm(1)1504 2097 y Fo(\000)1614 2058 y Fl(\031)r(i)p 1614 2074 67 4 v 1630 2131 a Fm(2)1691 2097 y Fi(\))1837 2025 y Fo(\000)23 b Fj(p)1986 2040 y Fm(2)2025 2025 y Fi(\()p Fj(\033)2118 2040 y Fm(1)2180 2025 y Fi(+)f(2)p Fj(\031)t(i)p Fi(\))2515 1957 y Fj(\037)2576 1972 y Fm(3)2615 1957 y Fi(\()p Fj(\033)2708 1972 y Fm(3)2748 1957 y Fo(j)p Fj(\033)2831 1972 y Fm(1)2871 1957 y Fi(\))p 2467 2002 490 4 v 2467 2097 a Fj(P)14 b Fi(\()p Fj(\033)2637 2112 y Fm(1)2698 2097 y Fi(+)2806 2058 y Fm(5)p Fl(\031)r(i)p 2806 2074 103 4 v 2840 2131 a Fm(2)2918 2097 y Fi(\))2966 1884 y Fn(\023)625 2309 y Fo(\000)23 b Fj(p)774 2324 y Fm(2)813 2309 y Fi(\()p Fj(\033)906 2324 y Fm(1)946 2309 y Fi(\))984 2168 y Fn(\022)1218 2242 y Fj(\037)1279 2257 y Fm(1)1319 2242 y Fi(\()p Fj(\033)1412 2257 y Fm(3)1452 2242 y Fo(j)p Fj(\033)1535 2257 y Fm(1)1596 2242 y Fo(\000)g Fj(\031)t(i)p Fi(\))p 1067 2286 910 4 v 1067 2381 a Fj(P)14 b Fi(\()p Fj(\033)1237 2396 y Fm(1)1299 2381 y Fo(\000)1408 2342 y Fl(\031)r(i)p 1408 2358 67 4 v 1424 2415 a Fm(2)1485 2381 y Fi(\))p Fj(P)g Fi(\()p Fj(\033)1693 2396 y Fm(1)1754 2381 y Fi(+)1862 2342 y Fl(\031)r(i)p 1862 2358 V 1878 2415 a Fm(2)1939 2381 y Fi(\))2009 2309 y(+)2286 2242 y Fj(\037)2347 2257 y Fm(3)2387 2242 y Fi(\()p Fj(\033)2480 2257 y Fm(3)2519 2242 y Fo(j)p Fj(\033)2602 2257 y Fm(1)2664 2242 y Fo(\000)23 b Fj(\031)t(i)p Fi(\))p 2117 2286 946 4 v 2117 2381 a Fj(P)14 b Fi(\()p Fj(\033)2287 2396 y Fm(1)2349 2381 y Fo(\000)2458 2342 y Fl(\031)r(i)p 2458 2358 67 4 v 2474 2415 a Fm(2)2535 2381 y Fi(\))p Fj(P)g Fi(\()p Fj(\033)2743 2396 y Fm(1)2804 2381 y Fi(+)2912 2342 y Fm(3)p Fl(\031)r(i)p 2912 2358 103 4 v 2946 2415 a Fm(2)3025 2381 y Fi(\))625 2593 y(+)877 2526 y Fj(\037)938 2541 y Fm(3)978 2526 y Fi(\()p Fj(\033)1071 2541 y Fm(3)1111 2526 y Fo(j)p Fj(\033)1194 2541 y Fm(1)1255 2526 y Fo(\000)23 b Fi(2)p Fj(\031)t(i)p Fi(\))p 733 2570 946 4 v 733 2665 a Fj(P)14 b Fi(\()p Fj(\033)903 2680 y Fm(1)964 2665 y Fo(\000)1074 2626 y Fm(3)p Fl(\031)r(i)p 1074 2642 103 4 v 1108 2700 a Fm(2)1186 2665 y Fi(\))p Fj(P)g Fi(\()p Fj(\033)1394 2680 y Fm(1)1456 2665 y Fi(+)1564 2626 y Fl(\031)r(i)p 1564 2642 67 4 v 1579 2700 a Fm(2)1640 2665 y Fi(\))1688 2453 y Fn(\023)631 2877 y Fi(=)27 b Fj(D)815 2892 y Fm(1)855 2737 y Fn(\022)928 2877 y Fj(p)977 2892 y Fm(2)1016 2877 y Fi(\()p Fj(\033)1109 2892 y Fm(1)1149 2877 y Fi(\))1197 2810 y Fj(\037)1258 2825 y Fm(3)1298 2810 y Fi(\()p Fj(\033)1391 2825 y Fm(3)1430 2810 y Fo(j)p Fj(\033)1513 2825 y Fm(1)1575 2810 y Fo(\000)c Fj(\031)t(i)p Fi(\))p 1197 2854 608 4 v 1273 2950 a Fj(P)14 b Fi(\()p Fj(\033)1443 2965 y Fm(1)1504 2950 y Fo(\000)1614 2910 y Fl(\031)r(i)p 1614 2927 67 4 v 1630 2984 a Fm(2)1691 2950 y Fi(\))1837 2877 y Fo(\000)23 b Fj(p)1986 2892 y Fm(2)2025 2877 y Fi(\()p Fj(\033)2118 2892 y Fm(1)2180 2877 y Fi(+)f(2)p Fj(\031)t(i)p Fi(\))2515 2810 y Fj(\037)2576 2825 y Fm(3)2615 2810 y Fi(\()p Fj(\033)2708 2825 y Fm(3)2748 2810 y Fo(j)p Fj(\033)2831 2825 y Fm(1)2871 2810 y Fi(\))p 2467 2854 490 4 v 2467 2950 a Fj(P)14 b Fi(\()p Fj(\033)2637 2965 y Fm(1)2698 2950 y Fi(+)2806 2910 y Fm(5)p Fl(\031)r(i)p 2806 2927 103 4 v 2840 2984 a Fm(2)2918 2950 y Fi(\))2966 2737 y Fn(\023)625 3162 y Fo(\000)23 b Fj(p)774 3177 y Fm(2)813 3162 y Fi(\()p Fj(\033)906 3177 y Fm(1)946 3162 y Fi(\))p Fj(\016)1031 3120 y Fk(\000)p Fm(1)1027 3186 y Fl(\033)1067 3195 y Fg(1)1125 3162 y Fj(R)1199 3177 y Fm(1)1239 3021 y Fn(\022)1312 3162 y Fj(D)1393 3177 y Fm(1)1433 3162 y Fj(f)1481 3177 y Fm(1)1520 3162 y Fi(\()p Fj(\033)1613 3177 y Fm(3)1653 3162 y Fo(j)p Fj(\033)1736 3177 y Fm(1)1776 3162 y Fo(j)p Fj(\033)1859 3177 y Fm(2)1898 3162 y Fi(\))f Fo(\000)h Fj(D)2139 3177 y Fm(2)2178 3162 y Fj(f)2226 3177 y Fm(1)2266 3162 y Fi(\()p Fj(\033)2359 3177 y Fm(3)2399 3162 y Fo(j)p Fj(\033)2482 3177 y Fm(2)2521 3162 y Fo(j)p Fj(\033)2604 3177 y Fm(1)2644 3162 y Fi(\))2682 3021 y Fn(\023)3888 3162 y Fy(\(104\))88 3432 y(The)32 b(last)g(equality)g(follo)n(ws)f(from)h(the)g(relation)g (\(94\).)54 b(Substituting)30 b(this)i(result)g(for)h(the)f(inte)o (gration)f(o)o(v)o(er)g Fj(\033)4063 3447 y Fm(2)88 3552 y Fy(into)23 b(\(103\))i(and)g(using)f(the)g(f)o(act)i(that)e(the)h (term)f Fj(D)1847 3567 y Fm(1)1887 3552 y Fi(\()p Fj(:)17 b(:)g(:)o Fi(\))25 b Fy(in)f(\(104\))h(does)f(not)h(contrib)n(ute)f (one)g(arri)n(v)o(es)g(at)756 3822 y Fo(F)828 3837 y Fm(2)895 3822 y Fi(=)998 3686 y Fn(Z)1054 3912 y Fm(\000)1098 3921 y Fg(1)1163 3754 y Fj(d\033)1269 3769 y Fm(1)p 1163 3799 146 4 v 1165 3890 a Fi(2)p Fj(\031)t(i)1335 3686 y Fn(Z)1390 3912 y Fm(\000)1434 3921 y Fg(3)1500 3754 y Fj(d\033)1606 3769 y Fm(3)p 1500 3799 V 1502 3890 a Fi(2)p Fj(\031)t(i)1655 3822 y(X)1736 3837 y Fm(1)1776 3822 y Fi(\()p Fj(\033)1869 3837 y Fm(1)1909 3822 y Fi(\))p Fj(p)1996 3837 y Fm(2)2035 3822 y Fi(\()p Fj(\033)2128 3837 y Fm(1)2168 3822 y Fi(\)\010\()p Fj(\033)2369 3837 y Fm(1)2409 3822 y Fi(\))p Fj(B)2521 3837 y Fm(34)2596 3822 y Fi(\()p Fj(\033)2689 3837 y Fm(3)2729 3822 y Fi(\))778 4099 y Fo(\001)e Fj(\016)875 4058 y Fk(\000)p Fm(1)871 4123 y Fl(\033)911 4132 y Fg(1)969 4099 y Fj(R)1043 4114 y Fm(1)1083 3958 y 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y(No)n(w)g(let)h(us)f(note)g(that)h(the)f(part)h(of)g(\(83\))1091 5158 y Fo(\000)p Fj(I)1211 5174 y Fm(\(1\))p Fl(;)p Fm(\(3\))p Fl(;)p Fm(\(24\))1583 5158 y Fi(+)d Fj(I)1724 5174 y Fm(\(1\))p Fl(;)p Fm(\(4\))p Fl(;)p Fm(\(23\))2095 5158 y Fi(+)g Fj(I)2236 5174 y Fm(\(2\))p Fl(;)p Fm(\(3\))p Fl(;)p Fm(\(14\))2608 5158 y Fo(\000)g Fj(I)2750 5174 y Fm(\(2\))p Fl(;)p Fm(\(4\))p Fl(;)p Fm(\(13\))88 5378 y Fy(is)i(already)h(sk)o(e)n(w-symmetric)e(w)-6 b(.r)h(.t.)29 b(the)c(transposition)e Fi(3)k Fo($)g Fi(4)p Fy(.)k(It)25 b(is)f(left)h(to)f(treat)h(the)g(v)o(ery)f(\002rst)h(term)g(in)f (\(83\))1899 5598 y Fj(I)1942 5614 y Fm(\(1\))p Fl(;)p Fm(\(2\))p Fl(;)p Fm(\(34\))2046 5871 y Fy(26)p eop %%Page: 27 27 27 26 bop 88 289 a Fy(Proceeding)24 b(in)g(a)g(similar)f(w)o(ay)i(as)f (we)h(treated)f(abo)o(v)o(e)f(the)h(term)g Fj(I)2419 304 y Fm(0)2483 289 y Fy(by)g(the)g(deri)n(v)n(ation)e(of)i(the)g (formula)g(\(97\))g(and)88 409 y(using)f(the)i(formula)f(\(85\))h(for)g (the)g(function)f Fj(h)1707 424 y Fm(1)1771 409 y Fy(we)h(can)h(obtain) 923 622 y Fj(I)966 637 y Fm(\(1\))p Fl(;)p Fm(\(2\))p Fl(;)p Fm(\(34\))1337 622 y Fi(+)c Fj(I)1478 637 y Fm(\(1\))p Fl(;)p Fm(\(2\))p Fl(;)p Fm(\(34\))1827 658 y Fk(j)1847 667 y Fg(3)p Fe($)p Fg(4)1999 622 y Fi(+)g Fo(F)2169 637 y Fm(2)2236 622 y Fi(=)27 b Fj(J)2393 637 y Fm(1)3888 622 y Fy(\(107\))88 843 y(where)387 1090 y Fj(J)441 1105 y Fm(1)508 1090 y Fi(=)612 954 y Fn(Z)667 1180 y Fm(\000)711 1189 y Fg(1)776 1023 y Fj(d\033)882 1038 y Fm(1)p 776 1067 146 4 v 778 1158 a Fi(2)p Fj(\031)t(i)949 954 y Fn(Z)1004 1180 y Fm(\000)1048 1189 y Fg(3)1113 1023 y Fj(d\033)1219 1038 y Fm(3)p 1113 1067 V 1115 1158 a Fi(2)p Fj(\031)t(i)1269 1090 y(X)1350 1105 y Fm(1)1389 1090 y Fi(\()p Fj(\033)1482 1105 y Fm(1)1522 1090 y Fi(\))p Fj(p)1609 1105 y Fm(2)1648 1090 y Fi(\()p Fj(\033)1741 1105 y Fm(1)1781 1090 y Fi(\)\010\()p Fj(\033)1982 1105 y Fm(1)2022 1090 y Fi(\))p Fj(B)2134 1105 y Fm(34)2209 1090 y Fi(\()p Fj(\033)2302 1105 y Fm(3)2342 1090 y Fi(\))p 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Fi(\()p Fj(\033)1027 1535 y Fm(1)1066 1520 y Fo(j)p Fj(\033)1149 1535 y Fm(3)1189 1520 y Fo(j)p Fj(\033)1272 1535 y Fm(4)1311 1520 y Fi(\))h(+)f Fj(D)1551 1535 y Fm(4)1590 1520 y Fj(f)1638 1535 y Fm(1)1678 1520 y Fi(\()p Fj(\033)1771 1535 y Fm(1)1810 1520 y Fo(j)p Fj(\033)1893 1535 y Fm(4)1933 1520 y Fo(j)p Fj(\033)2016 1535 y Fm(3)2055 1520 y Fi(\)])2120 1380 y Fn(\023)3888 1520 y Fy(\(108\))88 1784 y(Actually)-6 b(,)23 b(e)o(xpression)h(in)g(the)h(curled)f(brack)o (et)i(coincides)e(with)g(that)g(from)h(the)f(formula)h(\(88\).)30 b(Therefore)586 2043 y Fj(J)640 2058 y Fm(1)707 2043 y Fi(=)811 1907 y Fn(Z)866 2133 y Fm(\000)910 2142 y Fg(1)975 1975 y Fj(d\033)1081 1990 y Fm(1)p 975 2020 V 977 2111 a Fi(2)p Fj(\031)t(i)1148 1907 y Fn(Z)1203 2133 y Fm(\000)1247 2142 y Fg(3)1312 1975 y Fj(d\033)1418 1990 y Fm(3)p 1312 2020 V 1314 2111 a Fi(2)p Fj(\031)t(i)1468 2043 y(X)1549 2058 y Fm(1)1588 2043 y Fi(\()p Fj(\033)1681 2058 y Fm(1)1721 2043 y Fi(\))p Fj(p)1808 2058 y Fm(2)1847 2043 y 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Fj(I)2307 4030 y Fm(2)3888 4015 y Fy(\(110\))88 4228 y(where)j Fj(p)405 4243 y Fm(1)444 4228 y Fj(;)17 b(p)537 4243 y Fm(2)576 4228 y Fj(;)g(p)669 4243 y Fm(3)708 4228 y Fj(;)g(p)801 4243 y Fm(4)841 4228 y Fj(;)g(p)934 4243 y Fm(5)998 4228 y Fy(belong)24 b(to)g(the)h(set)f (of)h(polynomials)88 4348 y Fj(s)134 4363 y Fk(\000)p Fm(5)228 4348 y Fj(;)17 b(:)g(:)g(:)f(;)h(s)493 4363 y Fk(\000)p Fm(2)587 4348 y Fj(;)g(s)677 4363 y Fk(\000)p Fm(1)771 4348 y Fj(;)g(s)861 4363 y Fm(1)900 4348 y Fj(;)g(s)990 4363 y Fm(2)1029 4348 y Fj(;)g(:)g(:)g(:)f(;)h(s)1294 4363 y Fm(5)1333 4348 y Fy(.)234 4469 y(The)25 b(main)f(term)g Fj(I)888 4484 y Fm(0)953 4469 y Fy(corresponds)g(to)g(the)h(partitions) 1475 4682 y Fj(S)34 b Fi(=)p 1672 4601 66 4 v 27 w Fj(S)g Fi(=)27 b Fo(;)100 b Fj(T)41 b Fi(=)28 b(\(1)p Fj(;)17 b Fi(2)p Fj(;)g Fi(3)p Fj(;)g Fi(4)p Fj(;)g Fi(5\))626 4987 y Fj(I)669 5002 y Fm(0)736 4987 y Fo(\021)28 b Fj(I)884 5002 y Fk(;)p Fl(;)p Fk(;)p Fl(;)p Fm(\(1)p Fl(;)p Fm(2)p Fl(;)p Fm(3)p Fl(;)p Fm(4)p Fl(;)p Fm(5\))1335 4987 y Fi(=)642 5057 y Fn(Z)697 5283 y Fm(\000)741 5292 y Fg(1)807 5125 y Fj(d\033)913 5140 y Fm(1)p 807 5170 146 4 v 809 5261 a Fi(2)p Fj(\031)t(i)979 5193 y(:)17 b(:)g(:)1110 5057 y Fn(Z)1165 5283 y Fm(\000)1209 5292 y Fg(5)1275 5125 y Fj(d\033)1381 5140 y Fm(5)p 1275 5170 V 1277 5261 a Fi(2)p Fj(\031)t(i)1430 5193 y(X)1511 5208 y Fm(1)1551 5193 y Fi(\()p Fj(\033)1644 5208 y Fm(1)1684 5193 y Fi(\)\010\()p Fj(\033)1885 5208 y Fm(1)1925 5193 y Fi(\))p Fj(X)2044 5208 y Fm(2)2083 5193 y Fi(\()p Fj(\033)2176 5208 y Fm(2)2216 5193 y Fi(\)\010\()p Fj(\033)2417 5208 y Fm(2)2457 5193 y Fi(\))p Fj(X)2576 5208 y Fm(3)2615 5193 y Fi(\()p Fj(\033)2708 5208 y Fm(3)2748 5193 y Fi(\)\010\()p Fj(\033)2949 5208 y Fm(3)2989 5193 y Fi(\))648 5409 y Fo(\001)k Fj(X)778 5424 y Fm(4)818 5409 y Fi(\()p Fj(\033)911 5424 y Fm(4)951 5409 y Fi(\)\010\()p Fj(\033)1152 5424 y Fm(4)1192 5409 y Fi(\))p Fj(X)1311 5424 y Fm(5)1350 5409 y Fi(\()p Fj(\033)1443 5424 y Fm(5)1483 5409 y Fi(\)\010\()p Fj(\033)1684 5424 y Fm(5)1724 5409 y Fi(\))p Fj(h)1818 5424 y Fm(0)1858 5409 y Fi(\()p Fj(\033)1951 5424 y Fm(1)1991 5409 y Fj(;)c(\033)2090 5424 y Fm(2)2129 5409 y Fj(;)g(\033)2228 5424 y Fm(3)2268 5409 y Fj(;)g(\033)2367 5424 y Fm(4)2406 5409 y Fj(;)g(\033)2505 5424 y Fm(5)2545 5409 y Fi(\))1305 b Fy(\(111\))88 5622 y(where)25 b(as)g(abo)o(v)o(e)f(the)g(function)g Fj(h)1283 5637 y Fm(0)1323 5622 y Fi(\()p Fj(\033)1416 5637 y Fm(1)1456 5622 y Fj(;)17 b(:)g(:)g(:)e(;)i(\033)1729 5637 y Fm(5)1769 5622 y Fi(\))25 b Fy(is)f(de\002ned)h(by)g(the)g(formula)f(\(22\).)2046 5871 y(27)p eop %%Page: 28 28 28 27 bop 234 289 a Fy(F)o(or)25 b(the)f(ne)o(xt)g(term)h(with)f Fj(p)j Fi(=)h(1)d Fy(there)g(are)g(ten)g(contrib)n(utions)e (corresponding)h(to)g(ten)h(partitions)1462 509 y Fj(S)34 b Fi(=)27 b(\(1\))p 1883 429 66 4 v 99 w Fj(S)34 b Fi(=)27 b(\(2\))100 b Fj(T)41 b Fi(=)28 b(\(345\))1462 729 y Fj(S)34 b Fi(=)27 b(\(1\))p 1883 649 V 99 w Fj(S)34 b Fi(=)27 b(\(3\))100 b Fj(T)41 b Fi(=)28 b(\(245\))1462 903 y Fj(S)34 b Fi(=)27 b(\(1\))p 1883 823 V 99 w Fj(S)34 b Fi(=)27 b(\(4\))100 b Fj(T)41 b Fi(=)28 b(\(235\))1462 1077 y Fj(S)34 b Fi(=)27 b(\(1\))p 1883 997 V 99 w Fj(S)34 b Fi(=)27 b(\(5\))100 b Fj(T)41 b Fi(=)28 b(\(234\))1462 1252 y Fj(S)34 b Fi(=)27 b(\(2\))p 1883 1172 V 99 w Fj(S)34 b Fi(=)27 b(\(3\))100 b Fj(T)41 b Fi(=)28 b(\(145\))1462 1426 y Fj(S)34 b Fi(=)27 b(\(2\))p 1883 1346 V 99 w Fj(S)34 b Fi(=)27 b(\(4\))100 b Fj(T)41 b Fi(=)28 b(\(135\))1462 1600 y Fj(S)34 b Fi(=)27 b(\(2\))p 1883 1520 V 99 w Fj(S)34 b Fi(=)27 b(\(5\))100 b Fj(T)41 b Fi(=)28 b(\(134\))1462 1775 y Fj(S)34 b Fi(=)27 b(\(3\))p 1883 1695 V 99 w Fj(S)34 b Fi(=)27 b(\(4\))100 b Fj(T)41 b Fi(=)28 b(\(125\))1462 1949 y Fj(S)34 b Fi(=)27 b(\(3\))p 1883 1869 V 99 w Fj(S)34 b Fi(=)27 b(\(5\))100 b Fj(T)41 b Fi(=)28 b(\(124\))1462 2123 y Fj(S)34 b Fi(=)27 b(\(4\))p 1883 2043 V 99 w Fj(S)34 b Fi(=)27 b(\(5\))100 b Fj(T)41 b Fi(=)28 b(\(123\))88 2298 y Fy(and)c(according)h(to)f(the)h(formulae)g(\(25,26\))481 2518 y Fj(I)524 2533 y Fm(1)591 2518 y Fi(=)j Fj(I)738 2533 y Fm(\(1\))p Fl(;)p Fm(\(2\))p Fl(;)p Fm(\(345\))1144 2518 y Fo(\000)23 b Fj(I)1287 2533 y Fm(\(1\))p Fl(;)p Fm(\(3\))p Fl(;)p Fm(\(245\))1694 2518 y Fi(+)f Fj(I)1835 2533 y Fm(\(1\))p Fl(;)p Fm(\(4\))p Fl(;)p Fm(\(235\))2241 2518 y Fo(\000)h Fj(I)2384 2533 y Fm(\(1\))p Fl(;)p Fm(\(5\))p Fl(;)p Fm(\(234\))2791 2518 y Fi(+)f Fj(I)2932 2533 y Fm(\(2\))p Fl(;)p Fm(\(3\))p Fl(;)p Fm(\(145\))503 2663 y Fo(\000)h Fj(I)646 2679 y Fm(\(2\))p Fl(;)p Fm(\(4\))p Fl(;)p Fm(\(135\))1053 2663 y Fi(+)f Fj(I)1194 2679 y Fm(\(2\))p Fl(;)p Fm(\(5\))p Fl(;)p Fm(\(134\))1600 2663 y Fi(+)g Fj(I)1741 2679 y Fm(\(3\))p Fl(;)p Fm(\(4\))p Fl(;)p Fm(\(125\))2148 2663 y Fo(\000)g Fj(I)2290 2679 y Fm(\(3\))p Fl(;)p Fm(\(5\))p Fl(;)p Fm(\(124\))2697 2663 y Fi(+)g Fj(I)2838 2679 y Fm(\(4\))p Fl(;)p Fm(\(5\))p Fl(;)p Fm(\(123\))3888 2663 y Fy(\(112\))88 2883 y(where)381 3129 y Fj(I)424 3145 y Fm(\()p Fl(i)475 3154 y Fg(1)510 3145 y Fm(\))p Fl(;)p Fm(\()p Fl(i)608 3154 y Fg(2)643 3145 y Fm(\))p Fl(;)p Fm(\()p Fl(i)741 3154 y Fg(3)788 3145 y Fl(i)812 3154 y Fg(4)858 3145 y Fl(i)882 3154 y Fg(5)917 3145 y Fm(\))976 3129 y Fi(=)1079 2993 y Fn(Z)1135 3219 y Fm(\000)1179 3229 y Ff(i)1201 3244 y Fg(1)1271 3062 y Fj(d\033)1377 3077 y Fl(i)1401 3086 y Fg(1)p 1271 3106 169 4 v 1285 3197 a Fi(2)p Fj(\031)t(i)1466 2993 y Fn(Z)1521 3219 y Fm(\000)1565 3229 y Ff(i)1587 3244 y Fg(3)1657 3062 y Fj(d\033)1763 3077 y Fl(i)1787 3086 y Fg(3)p 1657 3106 V 1671 3197 a Fi(2)p Fj(\031)t(i)1853 2993 y Fn(Z)1908 3219 y Fm(\000)1952 3229 y Ff(i)1974 3244 y Fg(4)2044 3062 y Fj(d\033)2150 3077 y Fl(i)2174 3086 y Fg(4)p 2044 3106 V 2058 3197 a Fi(2)p Fj(\031)t(i)2239 2993 y Fn(Z)2294 3219 y Fm(\000)2338 3229 y Ff(i)2360 3244 y Fg(5)2430 3062 y Fj(d\033)2536 3077 y Fl(i)2560 3086 y Fg(5)p 2430 3106 V 2444 3197 a Fi(2)p Fj(\031)t(i)2609 3129 y(X)2690 3144 y Fl(i)2714 3153 y Fg(1)2753 3129 y Fi(\()p Fj(\033)2846 3144 y Fl(i)2870 3153 y Fg(1)2909 3129 y Fi(\))p Fj(p)2996 3144 y Fl(i)3020 3153 y Fg(2)3059 3129 y Fi(\()p Fj(\033)3152 3144 y Fl(i)3176 3153 y Fg(1)3215 3129 y Fi(\)\010\()p Fj(\033)3416 3144 y Fl(i)3440 3153 y Fg(1)3479 3129 y Fi(\))403 3362 y Fo(\001)g Fj(X)534 3377 y Fl(i)558 3386 y Fg(3)596 3362 y Fi(\()p Fj(\033)689 3377 y Fl(i)713 3386 y Fg(3)752 3362 y Fi(\)\010\()p Fj(\033)953 3377 y Fl(i)977 3386 y Fg(3)1017 3362 y Fi(\))p Fj(X)1136 3377 y Fl(i)1160 3386 y Fg(4)1199 3362 y Fi(\()p Fj(\033)1292 3377 y Fl(i)1316 3386 y Fg(4)1355 3362 y Fi(\)\010\()p Fj(\033)1556 3377 y Fl(i)1580 3386 y Fg(4)1619 3362 y Fi(\))p Fj(X)1738 3377 y Fl(i)1762 3386 y Fg(5)1801 3362 y Fi(\()p Fj(\033)1894 3377 y Fl(i)1918 3386 y Fg(5)1957 3362 y Fi(\)\010\()p Fj(\033)2158 3377 y Fl(i)2182 3386 y Fg(5)2222 3362 y Fi(\))p Fj(h)2316 3377 y Fm(1)2355 3362 y Fi(\()p Fj(\033)2448 3377 y Fl(i)2472 3386 y Fg(1)2511 3362 y Fo(j)p Fj(\033)2594 3377 y Fl(i)2618 3386 y Fg(3)2657 3362 y Fj(;)17 b(\033)2756 3377 y Fl(i)2780 3386 y Fg(4)2819 3362 y Fj(;)g(\033)2918 3377 y Fl(i)2942 3386 y Fg(5)2981 3362 y Fi(\))869 b Fy(\(113\))88 3582 y(and)24 b(due)h(to)g(the)f (formula)g(\(28\))446 3852 y Fj(h)502 3867 y Fm(1)542 3852 y Fi(\()p Fj(\033)635 3867 y Fm(1)675 3852 y Fo(j)p Fj(\033)758 3867 y Fm(3)797 3852 y Fj(;)17 b(\033)896 3867 y Fm(4)936 3852 y Fj(;)g(\033)1035 3867 y Fm(5)1074 3852 y Fi(\))28 b(=)f Fj(\016)1290 3810 y Fk(\000)p Fm(1)1286 3876 y Fl(\033)1326 3885 y Fg(1)1385 3711 y Fn(\022)1458 3852 y Fj(R)1532 3867 y Fm(1)1572 3852 y Fj(h)1628 3867 y Fm(0)1667 3852 y Fi(\()p Fj(\033)1760 3867 y Fm(1)1800 3852 y Fj(;)17 b(\033)1899 3867 y Fm(2)1939 3852 y Fj(;)g(\033)2038 3867 y Fm(3)2077 3852 y Fj(;)g(\033)2176 3867 y Fm(4)2216 3852 y Fj(;)g(\033)2315 3867 y Fm(5)2354 3852 y Fi(\))468 4124 y Fo(\000)23 b Fj(D)649 4139 y Fm(3)688 4124 y Fj(f)736 4139 y Fm(1)776 4124 y Fi(\()p Fj(\033)869 4139 y Fm(1)909 4124 y Fo(j)p Fj(\033)992 4139 y Fm(3)1031 4124 y Fo(j)p Fj(\033)1114 4139 y Fm(4)1154 4124 y Fj(;)17 b(\033)1253 4139 y Fm(5)1292 4124 y Fi(\))22 b(+)g Fj(D)1531 4139 y Fm(4)1571 4124 y Fj(f)1619 4139 y Fm(1)1658 4124 y Fi(\()p Fj(\033)1751 4139 y Fm(1)1791 4124 y Fo(j)p Fj(\033)1874 4139 y Fm(4)1914 4124 y Fo(j)p Fj(\033)1997 4139 y Fm(3)2036 4124 y Fj(;)17 b(\033)2135 4139 y Fm(5)2175 4124 y Fi(\))22 b Fo(\000)h Fj(D)2416 4139 y Fm(5)2455 4124 y Fj(f)2503 4139 y Fm(1)2543 4124 y Fi(\()p Fj(\033)2636 4139 y Fm(1)2675 4124 y Fo(j)p Fj(\033)2758 4139 y Fm(5)2798 4124 y Fo(j)p Fj(\033)2881 4139 y Fm(3)2921 4124 y Fj(;)17 b(\033)3020 4139 y Fm(4)3059 4124 y Fi(\))3097 3983 y Fn(\023)3888 4124 y Fy(\(114\))88 4401 y(Ob)o(viously)22 b(the)j(function)f Fj(h)1086 4416 y Fm(1)1125 4401 y Fi(\()p Fj(\033)1218 4416 y Fm(1)1258 4401 y Fo(j)p Fj(\033)1341 4416 y Fm(3)1381 4401 y Fj(;)17 b(\033)1480 4416 y Fm(4)1519 4401 y Fj(;)g(\033)1618 4416 y Fm(5)1658 4401 y Fi(\))25 b Fy(is)f(fully)g(anti-symmetric)f (for)i(the)g(v)n(ariables)f Fj(\033)3378 4416 y Fm(3)3418 4401 y Fj(;)17 b(\033)3517 4416 y Fm(4)3557 4401 y Fj(;)g(\033)3656 4416 y Fm(5)3695 4401 y Fy(.)234 4522 y(F)o(or)24 b(the)f(last)h(term)f (with)g Fj(p)28 b Fi(=)f(2)d Fy(there)g(are)h(\002fteen)f(contrib)n (utions)e(corresponding)h(to)g(all)g(possible)g(partitions)88 4642 y(that)h(satisfy)g(the)h(conditions)e(\(24\))1462 4862 y Fj(S)34 b Fi(=)27 b(\(12\))p 1932 4782 66 4 v 99 w Fj(S)34 b Fi(=)27 b(\(34\))99 b Fj(T)42 b Fi(=)27 b(\(5\))1462 5082 y Fj(S)34 b Fi(=)27 b(\(12\))p 1932 5002 V 99 w Fj(S)34 b Fi(=)27 b(\(43\))99 b Fj(T)42 b Fi(=)27 b(\(5\))1462 5256 y Fj(S)34 b Fi(=)27 b(\(12\))p 1932 5176 V 99 w Fj(S)34 b Fi(=)27 b(\(35\))99 b Fj(T)42 b Fi(=)27 b(\(4\))1462 5431 y Fj(S)34 b Fi(=)27 b(\(12\))p 1932 5351 V 99 w Fj(S)34 b Fi(=)27 b(\(53\))99 b Fj(T)42 b Fi(=)27 b(\(4\))1462 5605 y Fj(S)34 b Fi(=)27 b(\(12\))p 1932 5525 V 99 w Fj(S)34 b Fi(=)27 b(\(45\))99 b Fj(T)42 b Fi(=)27 b(\(3\))2046 5871 y Fy(28)p eop %%Page: 29 29 29 28 bop 1462 289 a Fj(S)34 b Fi(=)27 b(\(12\))p 1932 209 66 4 v 99 w Fj(S)34 b Fi(=)27 b(\(54\))99 b Fj(T)42 b Fi(=)27 b(\(3\))1462 460 y Fj(S)34 b Fi(=)27 b(\(13\))p 1932 380 V 99 w Fj(S)34 b Fi(=)27 b(\(24\))99 b Fj(T)42 b Fi(=)27 b(\(5\))1462 630 y Fj(S)34 b Fi(=)27 b(\(13\))p 1932 550 V 99 w Fj(S)34 b Fi(=)27 b(\(25\))99 b Fj(T)42 b Fi(=)27 b(\(4\))1462 801 y Fj(S)34 b Fi(=)27 b(\(13\))p 1932 721 V 99 w Fj(S)34 b Fi(=)27 b(\(45\))99 b Fj(T)42 b Fi(=)27 b(\(2\))1462 972 y Fj(S)34 b Fi(=)27 b(\(13\))p 1932 892 V 99 w Fj(S)34 b Fi(=)27 b(\(54\))99 b Fj(T)42 b Fi(=)27 b(\(2\))1462 1143 y Fj(S)34 b Fi(=)27 b(\(14\))p 1932 1063 V 99 w Fj(S)34 b Fi(=)27 b(\(25\))99 b Fj(T)42 b Fi(=)27 b(\(3\))1462 1314 y Fj(S)34 b Fi(=)27 b(\(14\))p 1932 1234 V 99 w Fj(S)34 b Fi(=)27 b(\(35\))99 b Fj(T)42 b Fi(=)27 b(\(2\))1462 1485 y Fj(S)34 b Fi(=)27 b(\(23\))p 1932 1405 V 99 w Fj(S)34 b Fi(=)27 b(\(45\))99 b Fj(T)42 b Fi(=)27 b(\(1\))1462 1655 y Fj(S)34 b Fi(=)27 b(\(23\))p 1932 1575 V 99 w Fj(S)34 b Fi(=)27 b(\(54\))99 b Fj(T)42 b Fi(=)27 b(\(1\))1462 1826 y Fj(S)34 b Fi(=)27 b(\(24\))p 1932 1746 V 99 w Fj(S)34 b Fi(=)27 b(\(35\))99 b Fj(T)42 b Fi(=)27 b(\(1\))88 1997 y Fy(and)455 2209 y Fj(I)498 2224 y Fm(2)566 2209 y Fi(=)g Fo(\000)p Fj(I)789 2224 y Fm(\(12\))p Fl(;)p Fm(\(34\))p Fl(;)p Fm(\(5\))1196 2209 y Fi(+)22 b Fj(I)1337 2224 y Fm(\(12\))p Fl(;)p Fm(\(43\))p Fl(;)p Fm(\(5\))1744 2209 y Fi(+)g Fj(I)1885 2224 y Fm(\(12\))p Fl(;)p Fm(\(35\))p Fl(;)p Fm(\(4\))2291 2209 y Fo(\000)h Fj(I)2434 2224 y Fm(\(12\))p Fl(;)p Fm(\(53\))p Fl(;)p Fm(\(4\))2841 2209 y Fo(\000)f Fj(I)2983 2224 y Fm(\(12\))p Fl(;)p Fm(\(45\))p Fl(;)p Fm(\(3\))478 2354 y Fi(+)g Fj(I)619 2370 y Fm(\(12\))p Fl(;)p Fm(\(54\))p Fl(;)p Fm(\(3\))1025 2354 y Fi(+)g Fj(I)1166 2370 y Fm(\(13\))p Fl(;)p Fm(\(24\))p Fl(;)p Fm(\(5\))1573 2354 y Fo(\000)g Fj(I)1715 2370 y Fm(\(13\))p Fl(;)p Fm(\(25\))p Fl(;)p Fm(\(4\))2122 2354 y Fi(+)g Fj(I)2263 2370 y Fm(\(13\))p Fl(;)p Fm(\(45\))p Fl(;)p Fm(\(2\))2670 2354 y Fo(\000)g Fj(I)2812 2370 y Fm(\(13\))p Fl(;)p Fm(\(54\))p Fl(;)p Fm(\(2\))478 2500 y Fi(+)g Fj(I)619 2515 y Fm(\(14\))p Fl(;)p Fm(\(25\))p Fl(;)p Fm(\(3\))1025 2500 y Fo(\000)h Fj(I)1168 2515 y Fm(\(14\))p Fl(;)p Fm(\(35\))p Fl(;)p Fm(\(2\))1574 2500 y Fo(\000)g Fj(I)1717 2515 y Fm(\(23\))p Fl(;)p Fm(\(45\))p Fl(;)p Fm(\(1\))2124 2500 y Fi(+)f Fj(I)2265 2515 y Fm(\(23\))p Fl(;)p Fm(\(54\))p Fl(;)p Fm(\(1\))2671 2500 y Fi(+)g Fj(I)2812 2515 y Fm(\(24\))p Fl(;)p Fm(\(35\))p Fl(;)p Fm(\(1\))3888 2500 y Fy(\(115\))88 2711 y(where)509 2949 y Fj(I)552 2965 y Fm(\()p Fl(i)603 2974 y Fg(1)650 2965 y Fl(i)674 2974 y Fg(2)709 2965 y Fm(\))p Fl(;)p Fm(\()p Fl(i)807 2974 y Fg(3)854 2965 y Fl(i)878 2974 y Fg(4)912 2965 y Fm(\))p Fl(;)p Fm(\()p Fl(i)1010 2974 y Fg(5)1046 2965 y Fm(\))1105 2949 y Fi(=)1208 2814 y Fn(Z)1264 3039 y Fm(\000)1308 3049 y Ff(i)1330 3064 y Fg(1)1399 2882 y Fj(d\033)1505 2897 y Fl(i)1529 2906 y Fg(1)p 1399 2926 169 4 v 1413 3017 a Fi(2)p Fj(\031)t(i)1595 2814 y Fn(Z)1650 3039 y Fm(\000)1694 3049 y Ff(i)1716 3064 y Fg(2)1786 2882 y Fj(d\033)1892 2897 y Fl(i)1916 2906 y Fg(2)p 1786 2926 V 1800 3017 a Fi(2)p Fj(\031)t(i)1981 2814 y Fn(Z)2037 3039 y Fm(\000)2081 3049 y Ff(i)2103 3064 y Fg(5)2173 2882 y Fj(d\033)2279 2897 y Fl(i)2303 2906 y Fg(5)p 2173 2926 V 2186 3017 a Fi(2)p Fj(\031)t(i)2351 2949 y(X)2432 2964 y Fl(i)2456 2973 y Fg(1)2495 2949 y Fi(\()p Fj(\033)2588 2964 y Fl(i)2612 2973 y Fg(1)2651 2949 y Fi(\))p Fj(p)2738 2964 y Fl(i)2762 2973 y Fg(3)2801 2949 y Fi(\()p Fj(\033)2894 2964 y Fl(i)2918 2973 y Fg(1)2957 2949 y Fi(\)\010\()p Fj(\033)3158 2964 y Fl(i)3182 2973 y Fg(1)3222 2949 y Fi(\))509 3182 y Fj(X)590 3197 y Fl(i)614 3206 y Fg(2)653 3182 y Fi(\()p Fj(\033)746 3197 y Fl(i)770 3206 y Fg(2)809 3182 y Fi(\))p Fj(p)896 3197 y Fl(i)920 3206 y Fg(4)959 3182 y Fi(\()p Fj(\033)1052 3197 y Fl(i)1076 3206 y Fg(2)1115 3182 y Fi(\)\010\()p Fj(\033)1316 3197 y Fl(i)1340 3206 y Fg(2)1380 3182 y Fi(\))p Fj(X)1499 3197 y Fl(i)1523 3206 y Fg(5)1561 3182 y Fi(\()p Fj(\033)1654 3197 y Fl(i)1678 3206 y Fg(5)1718 3182 y Fi(\)\010\()p Fj(\033)1919 3197 y Fl(i)1943 3206 y Fg(5)1982 3182 y Fi(\))p Fj(h)2076 3197 y Fm(2)2116 3182 y Fi(\()p Fj(\033)2209 3197 y Fl(i)2233 3206 y Fg(1)2272 3182 y Fj(;)17 b(\033)2371 3197 y Fl(i)2395 3206 y Fg(2)2434 3182 y Fo(j)p Fj(\033)2517 3197 y Fl(i)2541 3206 y Fg(5)2580 3182 y Fi(\))1270 b Fy(\(116\))88 3393 y(and)24 b(due)h(to)g(the)f(formula)g(\(28\))487 3655 y Fj(h)543 3670 y Fm(2)582 3655 y Fi(\()p Fj(\033)675 3670 y Fm(1)715 3655 y Fj(;)17 b(\033)814 3670 y Fm(3)854 3655 y Fo(j)p Fj(\033)937 3670 y Fm(5)976 3655 y Fi(\))28 b(=)f Fj(\016)1192 3614 y Fk(\000)p Fm(1)1188 3680 y Fl(\033)1228 3689 y Fg(1)1287 3655 y Fj(\016)1334 3614 y Fk(\000)p Fm(1)1330 3680 y Fl(\033)1370 3689 y Fg(3)1428 3515 y Fn(\022)1501 3655 y Fj(R)1575 3670 y Fm(1)1615 3655 y Fj(R)1689 3670 y Fm(3)1729 3655 y Fj(h)1785 3670 y Fm(0)1824 3655 y Fi(\()p Fj(\033)1917 3670 y Fm(1)1957 3655 y Fj(;)17 b(\033)2056 3670 y Fm(2)2095 3655 y Fj(;)g(\033)2194 3670 y Fm(3)2234 3655 y Fj(;)g(\033)2333 3670 y Fm(4)2373 3655 y Fj(;)g(\033)2472 3670 y Fm(5)2511 3655 y Fi(\))509 3867 y Fo(\000)22 b Fj(R)682 3882 y Fm(1)722 3867 y Fi([)p Fj(D)830 3882 y Fm(1)870 3867 y Fj(f)918 3882 y Fm(1)957 3867 y Fi(\()p Fj(\033)1050 3882 y Fm(3)1090 3867 y Fo(j)p Fj(\033)1173 3882 y Fm(1)1212 3867 y Fo(j)p Fj(\033)1295 3882 y Fm(2)1335 3867 y Fj(;)17 b(\033)1434 3882 y Fm(5)1474 3867 y Fi(\))22 b Fo(\000)g Fj(D)1714 3882 y Fm(2)1754 3867 y Fj(f)1802 3882 y Fm(1)1841 3867 y Fi(\()p Fj(\033)1934 3882 y Fm(3)1974 3867 y Fo(j)p Fj(\033)2057 3882 y Fm(2)2097 3867 y Fo(j)p Fj(\033)2180 3882 y Fm(1)2219 3867 y Fj(;)17 b(\033)2318 3882 y Fm(5)2358 3867 y Fi(\))22 b(+)g Fj(D)2597 3882 y Fm(5)2636 3867 y Fj(f)2684 3882 y Fm(1)2724 3867 y Fi(\()p Fj(\033)2817 3882 y Fm(3)2857 3867 y Fo(j)p Fj(\033)2940 3882 y Fm(5)2979 3867 y Fo(j)p Fj(\033)3062 3882 y Fm(1)3102 3867 y Fj(;)17 b(\033)3201 3882 y Fm(2)3240 3867 y Fi(\)])509 4013 y Fo(\000)22 b Fj(R)682 4028 y Fm(3)722 4013 y Fi([)p Fj(D)830 4028 y Fm(3)870 4013 y Fj(f)918 4028 y Fm(1)957 4013 y Fi(\()p Fj(\033)1050 4028 y Fm(1)1090 4013 y Fo(j)p Fj(\033)1173 4028 y Fm(3)1212 4013 y Fo(j)p Fj(\033)1295 4028 y Fm(4)1335 4013 y Fj(;)17 b(\033)1434 4028 y Fm(5)1474 4013 y Fi(\))22 b Fo(\000)g Fj(D)1714 4028 y Fm(4)1754 4013 y Fj(f)1802 4028 y Fm(1)1841 4013 y Fi(\()p Fj(\033)1934 4028 y Fm(1)1974 4013 y Fo(j)p Fj(\033)2057 4028 y Fm(4)2097 4013 y Fo(j)p Fj(\033)2180 4028 y Fm(3)2219 4013 y Fj(;)17 b(\033)2318 4028 y Fm(5)2358 4013 y Fi(\))22 b(+)g Fj(D)2597 4028 y Fm(5)2636 4013 y Fj(f)2684 4028 y Fm(1)2724 4013 y Fi(\()p Fj(\033)2817 4028 y Fm(1)2857 4013 y Fo(j)p Fj(\033)2940 4028 y Fm(5)2979 4013 y Fo(j)p Fj(\033)3062 4028 y Fm(3)3102 4013 y Fj(;)17 b(\033)3201 4028 y Fm(4)3240 4013 y Fi(\)])509 4227 y(+)22 b Fj(D)688 4242 y Fm(5)727 4227 y Fj(f)775 4242 y Fm(2)815 4227 y Fi(\()p Fj(\033)908 4242 y Fm(1)947 4227 y Fj(;)17 b(\033)1046 4242 y Fm(3)1086 4227 y Fo(j)p Fj(\033)1169 4242 y Fm(5)1209 4227 y Fi(\))1247 4086 y Fn(\023)3888 4227 y Fy(\(117\))88 4496 y(Lik)o(e)24 b(in)h(the)f(case)i Fj(n)h Fi(=)h(5)d Fy(the)f(formula)h(for)g Fj(h)1687 4511 y Fm(2)1726 4496 y Fi(\()p Fj(\033)1819 4511 y Fm(1)1859 4496 y Fj(;)17 b(\033)1958 4511 y Fm(3)1998 4496 y Fo(j)p Fj(\033)2081 4511 y Fm(5)2120 4496 y Fi(\))25 b Fy(is)f(symmetric)g(w) -6 b(.r)h(.t.)29 b(the)c(transposition)e Fj(\033)3688 4511 y Fm(1)3755 4496 y Fo($)28 b Fj(\033)3938 4511 y Fm(3)3977 4496 y Fy(.)234 4616 y(In)36 b(order)f(to)h(de\002ne)g(the)f (functions)g Fj(f)1603 4631 y Fm(1)1678 4616 y Fy(and)h Fj(f)1906 4631 y Fm(2)1981 4616 y Fy(we)g(need)f(some)g(residues)h (which,)h(in)f(principle,)h(may)e(be)88 4736 y(computed)23 b(e)o(xplicitly)g(using)h(the)h(formula)f(\(22\))1046 4948 y(Res)1195 4963 y Fl(\033)1235 4972 y Fg(2)1270 4963 y Fm(=)p Fl(\033)1365 4972 y Fg(1)1400 4963 y Fm(+)p Fl(\031)r(i)1526 4948 y Fj(h)1582 4963 y Fm(0)1622 4948 y Fi(\()p Fj(\033)1715 4963 y Fm(1)1755 4948 y Fj(;)17 b(\033)1854 4963 y Fm(2)1893 4948 y Fj(;)g(\033)1992 4963 y Fm(3)2032 4948 y Fj(;)g(\033)2131 4963 y Fm(4)2170 4948 y Fj(;)g(\033)2269 4963 y Fm(5)2309 4948 y Fi(\))28 b(=)f Fj(V)2535 4963 y Fm(1)2574 4948 y Fi(\()p Fj(\033)2667 4963 y Fm(1)2707 4948 y Fo(j)p Fj(\033)2790 4963 y Fm(3)2830 4948 y Fj(;)17 b(\033)2929 4963 y Fm(4)2968 4948 y Fj(;)g(\033)3067 4963 y Fm(5)3107 4948 y Fi(\))939 5160 y Fy(Res)1088 5175 y Fl(\033)1128 5184 y Fg(2)1163 5175 y Fm(=)p Fl(\033)1258 5184 y Fg(1)1293 5175 y Fk(\000)p Fl(\031)r(i)1419 5160 y Fj(h)1475 5175 y Fm(0)1515 5160 y Fi(\()p Fj(\033)1608 5175 y Fm(1)1648 5160 y Fj(;)g(\033)1747 5175 y Fm(2)1786 5160 y Fj(;)g(\033)1885 5175 y Fm(3)1925 5160 y Fj(;)g(\033)2024 5175 y Fm(4)2063 5160 y Fj(;)g(\033)2162 5175 y Fm(5)2202 5160 y Fi(\))28 b(=)f Fj(V)2428 5175 y Fm(1)2467 5160 y Fi(\()p Fj(\033)2560 5175 y Fm(1)2622 5160 y Fo(\000)c Fj(\031)t(i)p Fo(j)p Fj(\033)2897 5175 y Fm(3)2937 5160 y Fj(;)17 b(\033)3036 5175 y Fm(4)3075 5160 y Fj(;)g(\033)3174 5175 y Fm(5)3214 5160 y Fi(\))1175 5451 y Fy(Res)1324 5466 y Fl(\033)1364 5475 y Fg(3)1399 5466 y Fm(=)p Fl(\033)1494 5475 y Fg(1)1529 5466 y Fm(+2)p Fl(\031)r(i)1690 5451 y Fj(V)1747 5466 y Fm(1)1786 5451 y Fi(\()p Fj(\033)1879 5466 y Fm(1)1919 5451 y Fo(j)p Fj(\033)2002 5466 y Fm(3)2042 5451 y Fj(;)g(\033)2141 5466 y Fm(4)2180 5451 y Fj(;)g(\033)2279 5466 y Fm(5)2319 5451 y Fi(\))28 b(=)f Fj(V)2545 5466 y Fm(2)2584 5451 y Fi(\()p Fj(\033)2677 5466 y Fm(1)2717 5451 y Fo(j)p Fj(\033)2800 5466 y Fm(4)2840 5451 y Fj(;)17 b(\033)2939 5466 y Fm(5)2978 5451 y Fi(\))1047 5622 y Fy(Res)1196 5637 y Fl(\033)1236 5646 y Fg(3)1271 5637 y Fm(=)p Fl(\033)1366 5646 y Fg(1)1401 5637 y Fk(\000)p Fl(\031)r(i)1527 5622 y Fj(V)1584 5637 y Fm(1)1623 5622 y Fi(\()p Fj(\033)1716 5637 y Fm(1)1756 5622 y Fo(j)p Fj(\033)1839 5637 y Fm(3)1878 5622 y Fj(;)g(\033)1977 5637 y Fm(4)2017 5622 y Fj(;)g(\033)2116 5637 y Fm(5)2156 5622 y Fi(\))27 b(=)h Fo(\000)p Fj(V)2459 5637 y Fm(2)2498 5622 y Fi(\()p Fj(\033)2591 5637 y Fm(1)2653 5622 y Fo(\000)23 b Fj(\031)t(i)p Fo(j)p Fj(\033)2928 5637 y Fm(4)2968 5622 y Fj(;)17 b(\033)3067 5637 y Fm(5)3106 5622 y Fi(\))2046 5871 y Fy(29)p eop %%Page: 30 30 30 29 bop 1313 409 a Fy(Res)1463 424 y Fl(\033)1503 433 y Fg(4)1538 424 y Fm(=)p Fl(\033)1633 433 y Fg(1)1667 424 y Fm(+3)p Fl(\031)r(i)1829 409 y Fj(V)1886 424 y Fm(2)1925 409 y Fi(\()p Fj(\033)2018 424 y Fm(1)2058 409 y Fo(j)p Fj(\033)2141 424 y Fm(4)2180 409 y Fj(;)17 b(\033)2279 424 y Fm(5)2319 409 y Fi(\))28 b(=)f Fj(V)2545 424 y Fm(3)2584 409 y Fi(\()p Fj(\033)2677 424 y Fm(1)2717 409 y Fo(j)p Fj(\033)2800 424 y Fm(5)2840 409 y Fi(\))1303 583 y Fy(Res)1453 598 y Fl(\033)1493 607 y Fg(4)1528 598 y Fm(=)p Fl(\033)1623 607 y Fg(1)1658 598 y Fm(+)p Fl(\031)r(i)1784 583 y Fj(V)1841 598 y Fm(2)1880 583 y Fi(\()p Fj(\033)1973 598 y Fm(1)2013 583 y Fo(j)p Fj(\033)2096 598 y Fm(4)2135 583 y Fj(;)17 b(\033)2234 598 y Fm(5)2274 583 y Fi(\))27 b(=)h Fj(V)2500 598 y Fm(2)p Fl(;)p Fm(1)2594 583 y Fi(\()p Fj(\033)2687 598 y Fm(1)2727 583 y Fo(j)p Fj(\033)2810 598 y Fm(5)2850 583 y Fi(\))1224 758 y Fy(Res)1373 773 y Fl(\033)1413 782 y Fg(4)1448 773 y Fm(=)p Fl(\033)1543 782 y Fg(1)1578 773 y Fk(\000)p Fl(\031)r(i)1704 758 y Fj(V)1761 773 y Fm(2)1800 758 y Fi(\()p Fj(\033)1893 773 y Fm(1)1933 758 y Fo(j)p Fj(\033)2016 773 y Fm(4)2056 758 y Fj(;)17 b(\033)2155 773 y Fm(5)2194 758 y Fi(\))28 b(=)f Fj(V)2420 773 y Fm(3)2460 758 y Fi(\()p Fj(\033)2553 773 y Fm(1)2615 758 y Fo(\000)22 b Fj(\031)t(i)p Fo(j)p Fj(\033)2889 773 y Fm(5)2929 758 y Fi(\))1331 932 y Fy(Res)1480 947 y Fl(\033)1520 956 y Fg(5)1555 947 y Fm(=)p Fl(\033)1650 956 y Fg(4)1685 947 y Fm(+)p Fl(\031)r(i)1811 932 y Fj(V)1868 947 y Fm(2)1907 932 y Fi(\()p Fj(\033)2000 947 y Fm(1)2040 932 y Fo(j)p Fj(\033)2123 947 y Fm(4)2163 932 y Fj(;)17 b(\033)2262 947 y Fm(5)2301 932 y Fi(\))28 b(=)2485 907 y(~)2470 932 y Fj(V)2527 947 y Fm(2)2567 932 y Fi(\()p Fj(\033)2660 947 y Fm(1)2700 932 y Fo(j)p Fj(\033)2783 947 y Fm(4)2822 932 y Fi(\))1444 1227 y Fy(Res)1593 1242 y Fl(\033)1633 1251 y Fg(5)1668 1242 y Fm(=)p Fl(\033)1763 1251 y Fg(1)1798 1242 y Fm(+4)p Fl(\031)r(i)1959 1227 y Fj(V)2016 1242 y Fm(3)2056 1227 y Fi(\()p Fj(\033)2149 1242 y Fm(1)2188 1227 y Fo(j)p Fj(\033)2271 1242 y Fm(5)2311 1227 y Fi(\))f(=)h Fj(V)2537 1242 y Fm(4)2576 1227 y Fi(\()p Fj(\033)2669 1242 y Fm(1)2709 1227 y Fi(\))1416 1401 y Fy(Res)1566 1416 y Fl(\033)1606 1425 y Fg(5)1641 1416 y Fm(=)p Fl(\033)1736 1425 y Fg(1)1771 1416 y Fm(+2)p Fl(\031)r(i)1932 1401 y Fj(V)1989 1416 y Fm(3)2028 1401 y Fi(\()p Fj(\033)2121 1416 y Fm(1)2161 1401 y Fo(j)p Fj(\033)2244 1416 y Fm(5)2283 1401 y Fi(\))g(=)g Fj(V)2510 1416 y Fm(3)p Fl(;)p Fm(2)2604 1401 y Fi(\()p Fj(\033)2697 1416 y Fm(1)2737 1401 y Fi(\))1434 1576 y Fy(Res)1583 1591 y Fl(\033)1623 1600 y Fg(5)1658 1591 y Fm(=)p Fl(\033)1753 1600 y Fg(1)1788 1591 y Fm(+)p Fl(\031)r(i)1914 1576 y Fj(V)1971 1591 y Fm(3)2010 1576 y Fi(\()p Fj(\033)2103 1591 y Fm(1)2143 1576 y Fo(j)p Fj(\033)2226 1591 y Fm(5)2266 1576 y Fi(\))f(=)h Fj(V)2492 1591 y Fm(3)p Fl(;)p Fm(1)2586 1576 y Fi(\()p Fj(\033)2679 1591 y Fm(1)2719 1576 y Fi(\))1316 1750 y Fy(Res)1465 1765 y Fl(\033)1505 1774 y Fg(5)1540 1765 y Fm(=)p Fl(\033)1635 1774 y Fg(1)1670 1765 y Fk(\000)p Fl(\031)r(i)1796 1750 y Fj(V)1853 1765 y Fm(3)1892 1750 y Fi(\()p Fj(\033)1985 1765 y Fm(1)2025 1750 y Fo(j)p Fj(\033)2108 1765 y Fm(5)2148 1750 y Fi(\))f(=)h Fo(\000)p Fj(V)2451 1765 y Fm(4)2490 1750 y Fi(\()p Fj(\033)2583 1765 y Fm(1)2645 1750 y Fo(\000)23 b Fj(\031)t(i)p Fi(\))1444 2045 y Fy(Res)1593 2060 y Fl(\033)1633 2069 y Fg(4)1668 2060 y Fm(=)p Fl(\033)1763 2069 y Fg(3)1798 2060 y Fm(+3)p Fl(\031)r(i)1974 2019 y Fi(~)1959 2045 y Fj(V)2016 2060 y Fm(2)2056 2045 y Fi(\()p Fj(\033)2149 2060 y Fm(3)2188 2045 y Fo(j)p Fj(\033)2271 2060 y Fm(4)2311 2045 y Fi(\))k(=)h Fj(V)2537 2060 y Fm(4)2576 2045 y Fi(\()p Fj(\033)2669 2060 y Fm(3)2709 2045 y Fi(\))1416 2219 y Fy(Res)1566 2234 y Fl(\033)1606 2243 y Fg(4)1641 2234 y Fm(=)p Fl(\033)1736 2243 y Fg(3)1771 2234 y Fm(+2)p Fl(\031)r(i)1947 2194 y Fi(~)1932 2219 y Fj(V)1989 2234 y Fm(2)2028 2219 y Fi(\()p Fj(\033)2121 2234 y Fm(3)2161 2219 y Fo(j)p Fj(\033)2244 2234 y Fm(4)2283 2219 y Fi(\))g(=)g Fj(V)2510 2234 y Fm(3)p Fl(;)p Fm(2)2604 2219 y Fi(\()p Fj(\033)2697 2234 y Fm(3)2737 2219 y Fi(\))1434 2393 y Fy(Res)1583 2408 y Fl(\033)1623 2417 y Fg(4)1658 2408 y Fm(=)p Fl(\033)1753 2417 y Fg(3)1788 2408 y Fm(+)p Fl(\031)r(i)1929 2368 y Fi(~)1914 2393 y Fj(V)1971 2408 y Fm(2)2010 2393 y Fi(\()p Fj(\033)2103 2408 y Fm(3)2143 2393 y Fo(j)p Fj(\033)2226 2408 y Fm(4)2266 2393 y Fi(\))f(=)2450 2368 y(~)2435 2393 y Fj(V)2492 2408 y Fm(2)p Fl(;)p Fm(1)2586 2393 y Fi(\()p Fj(\033)2679 2408 y Fm(3)2719 2393 y Fi(\))1495 2568 y Fy(Res)1644 2583 y Fl(\033)1684 2592 y Fg(4)1719 2583 y Fm(=)p Fl(\033)1814 2592 y Fg(3)1868 2542 y Fi(~)1853 2568 y Fj(V)1910 2583 y Fm(2)1950 2568 y Fi(\()p Fj(\033)2043 2583 y Fm(3)2082 2568 y Fo(j)p Fj(\033)2165 2583 y Fm(4)2205 2568 y Fi(\))g(=)2389 2542 y(~)2374 2568 y Fj(V)2431 2583 y Fm(2)p Fl(;)p Fm(0)2525 2568 y Fi(\()p Fj(\033)2618 2583 y Fm(3)2658 2568 y Fi(\))1327 2742 y Fy(Res)1476 2757 y Fl(\033)1516 2766 y Fg(4)1551 2757 y Fm(=)p Fl(\033)1646 2766 y Fg(3)1681 2757 y Fk(\000)p Fl(\031)r(i)1822 2717 y Fi(~)1807 2742 y Fj(V)1864 2757 y Fm(2)1903 2742 y Fi(\()p Fj(\033)1996 2757 y Fm(3)2036 2742 y Fo(j)p Fj(\033)2119 2757 y Fm(4)2159 2742 y Fi(\))g(=)h Fj(V)2385 2757 y Fm(3)p Fl(;)p Fm(1)2479 2742 y Fi(\()p Fj(\033)2572 2757 y Fm(3)2634 2742 y Fo(\000)23 b Fj(\031)t(i)p Fi(\))1274 2916 y Fy(Res)1423 2931 y Fl(\033)1463 2940 y Fg(4)1498 2931 y Fm(=)p Fl(\033)1593 2940 y Fg(3)1628 2931 y Fk(\000)p Fm(2)p Fl(\031)r(i)1804 2891 y Fi(~)1789 2916 y Fj(V)1846 2931 y Fm(2)1885 2916 y Fi(\()p Fj(\033)1978 2931 y Fm(3)2018 2916 y Fo(j)p Fj(\033)2101 2931 y Fm(4)2141 2916 y Fi(\))k(=)h Fo(\000)p Fj(V)2444 2931 y Fm(4)2484 2916 y Fi(\()p Fj(\033)2577 2931 y Fm(3)2639 2916 y Fo(\000)22 b Fi(2)p Fj(\031)t(i)p Fi(\))234 3091 y Fy(The)j(result)f(for)h(the)g(function)f Fj(f)1353 3106 y Fm(1)1417 3091 y Fy(looks)g(as)h(follo)n(ws:)391 3360 y Fj(f)439 3375 y Fm(1)479 3360 y Fi(\()p Fj(\033)572 3375 y Fm(1)611 3360 y Fo(j)p Fj(\033)694 3375 y Fm(3)734 3360 y Fo(j)p Fj(\033)817 3375 y Fm(4)857 3360 y Fj(;)17 b(\033)956 3375 y Fm(5)995 3360 y Fi(\))28 b(=)1174 3293 y Fj(V)1231 3308 y Fm(2)1271 3293 y Fi(\()p Fj(\033)1364 3308 y Fm(1)1403 3293 y Fo(j)p Fj(\033)1486 3308 y Fm(4)1526 3293 y Fj(;)17 b(\033)1625 3308 y Fm(5)1665 3293 y Fi(\))p 1174 3337 529 4 v 1373 3428 a Fj(\033)1428 3443 y Fm(13)413 3608 y Fo(\000)23 b Fi(\()601 3541 y(1)p 561 3585 131 4 v 561 3677 a Fj(\033)616 3692 y Fm(13)723 3608 y Fo(\000)980 3541 y Fi(1)p 833 3585 345 4 v 833 3677 a Fj(\033)888 3692 y Fm(13)985 3677 y Fo(\000)g Fj(\031)t(i)1187 3608 y Fi(\))p Fj(p)p Fi(\()p Fj(\033)1367 3623 y Fm(3)1406 3608 y Fo(j)p Fj(\033)1489 3623 y Fm(4)1529 3608 y Fj(;)17 b(\033)1628 3623 y Fm(5)1668 3608 y Fi(\))22 b Fo(\000)g Fi(\()2023 3541 y(1)p 1875 3585 V 1875 3677 a Fj(\033)1930 3692 y Fm(13)2027 3677 y Fo(\000)h Fj(\031)t(i)2251 3608 y Fo(\000)2533 3541 y Fi(1)p 2361 3585 393 4 v 2361 3677 a Fj(\033)2416 3692 y Fm(13)2513 3677 y Fo(\000)g Fi(2)p Fj(\031)t(i)2764 3608 y Fi(\))p Fj(q)t Fi(\()p Fj(\033)2942 3623 y Fm(3)2981 3608 y Fo(j)p Fj(\033)3064 3623 y Fm(4)3104 3608 y Fj(;)17 b(\033)3203 3623 y Fm(5)3243 3608 y Fi(\))607 b Fy(\(118\))88 3867 y(where)716 4119 y Fj(p)p Fi(\()p Fj(\033)858 4134 y Fm(3)897 4119 y Fo(j)p Fj(\033)980 4134 y Fm(4)1020 4119 y Fj(;)17 b(\033)1119 4134 y Fm(5)1158 4119 y Fi(\))28 b(=)1338 4051 y Fj(V)1395 4066 y Fm(3)1435 4051 y Fi(\()p Fj(\033)1528 4066 y Fm(3)1567 4051 y Fo(j)p Fj(\033)1650 4066 y Fm(4)1690 4051 y Fi(\))p 1338 4096 392 4 v 1338 4187 a Fj(\033)1393 4202 y Fm(35)1490 4187 y Fi(+)22 b(3)p Fj(\031)t(i)1761 4119 y Fo(\000)1871 4051 y Fj(V)1928 4066 y Fm(3)1968 4051 y Fi(\()p Fj(\033)2061 4066 y Fm(3)2100 4051 y Fo(j)p Fj(\033)2183 4066 y Fm(5)2223 4051 y Fi(\))p 1871 4096 V 1871 4187 a Fj(\033)1926 4202 y Fm(34)2023 4187 y Fi(+)g(3)p Fj(\031)t(i)2272 4119 y Fo(\000)738 4377 y(\000)1157 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V 1874 4714 a(\()p Fj(\033)1967 4729 y Fm(34)2065 4714 y Fi(+)f(2)p Fj(\031)t(i)p Fi(\)\()p Fj(\033)2435 4729 y Fm(54)2532 4714 y Fo(\000)g Fj(\031)t(i)p Fi(\))3888 4646 y Fy(\(119\))88 4915 y(and)434 5166 y Fj(q)t Fi(\()p Fj(\033)574 5181 y Fm(3)613 5166 y Fo(j)p Fj(\033)696 5181 y Fm(4)736 5166 y Fj(;)17 b(\033)835 5181 y Fm(5)874 5166 y Fi(\))28 b(=)1364 5099 y Fj(V)1421 5114 y Fm(4)1460 5099 y Fi(\()p Fj(\033)1553 5114 y Fm(3)1593 5099 y Fi(\))p 1054 5143 V 1054 5235 a(\()p Fj(\033)1147 5250 y Fm(35)1244 5235 y Fi(+)22 b(3)p Fj(\031)t(i)p Fi(\)\()p Fj(\033)1614 5250 y Fm(45)1711 5235 y Fo(\000)h Fj(\031)t(i)p Fi(\))1973 5166 y Fo(\000)2392 5099 y Fj(V)2449 5114 y Fm(4)2489 5099 y Fi(\()p Fj(\033)2582 5114 y Fm(3)2622 5099 y Fi(\))p 2082 5143 V 2082 5235 a(\()p Fj(\033)2175 5250 y Fm(34)2273 5235 y Fi(+)f(3)p Fj(\031)t(i)p Fi(\)\()p Fj(\033)2643 5250 y Fm(54)2740 5235 y Fo(\000)g Fj(\031)t(i)p Fi(\))2980 5166 y Fo(\000)456 5449 y(\000)851 5382 y Fj(V)908 5397 y Fm(4)947 5382 y 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Fi(\))p 2912 5426 490 4 v 2912 5521 a Fj(P)g Fi(\()p Fj(\033)3082 5536 y Fm(3)3143 5521 y Fi(+)3251 5482 y Fm(7)p Fl(\031)r(i)p 3251 5498 103 4 v 3284 5556 a Fm(2)3363 5521 y Fi(\))3888 5449 y Fy(\(120\))2046 5871 y(30)p eop %%Page: 31 31 31 30 bop 234 289 a Fy(The)25 b(answer)g(for)g(the)f(function)g Fj(f)1413 304 y Fm(2)1478 289 y Fy(is)462 558 y Fj(f)510 573 y Fm(2)549 558 y Fi(\()p Fj(\033)642 573 y Fm(1)682 558 y Fj(;)17 b(\033)781 573 y Fm(3)821 558 y Fo(j)p Fj(\033)904 573 y Fm(5)943 558 y Fi(\))28 b(=)1654 491 y Fj(V)1711 506 y Fm(4)1750 491 y Fi(\()p Fj(\033)1843 506 y Fm(5)1883 491 y Fi(\))p 1122 535 1330 4 v 1122 630 a(\()p Fj(\033)1215 645 y Fm(15)1313 630 y Fo(\000)22 b Fj(\031)t(i)p Fi(\)\()p Fj(\033)1635 645 y Fm(35)1733 630 y Fo(\000)g Fj(\031)t(i)p Fi(\))p Fj(P)14 b Fi(\()p Fj(\033)2132 645 y Fm(5)2194 630 y Fi(+)2302 591 y Fm(7)p Fl(\031)r(i)p 2302 607 103 4 v 2335 665 a Fm(2)2414 630 y Fi(\))2484 558 y Fo(\000)2836 491 y Fj(V)2893 506 y Fm(4)2932 491 y Fi(\()p 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y Fi(+)2627 873 y Fm(5)p Fl(\031)r(i)p 2627 889 103 4 v 2661 947 a Fm(2)2740 912 y Fi(\))484 1138 y(+)803 1046 y(~)788 1071 y Fj(V)845 1086 y Fm(2)p Fl(;)p Fm(0)939 1071 y Fi(\()p Fj(\033)1032 1086 y Fm(5)1072 1071 y Fi(\))p 592 1115 715 4 v 592 1211 a Fj(\033)647 1226 y Fm(15)722 1211 y Fj(\033)777 1226 y Fm(35)852 1211 y Fj(P)g Fi(\()p Fj(\033)1022 1226 y Fm(5)1084 1211 y Fi(+)1192 1171 y Fl(\031)r(i)p 1192 1188 67 4 v 1207 1245 a Fm(2)1268 1211 y Fi(\))1339 1138 y(+)1675 1046 y(~)1660 1071 y Fj(V)1717 1086 y Fm(2)p Fl(;)p Fm(1)1812 1071 y Fi(\()p Fj(\033)1905 1086 y Fm(5)1944 1071 y Fi(\))p 1446 1115 750 4 v 1446 1211 a Fj(\033)1501 1226 y Fm(15)1577 1211 y Fj(\033)1632 1226 y Fm(35)1707 1211 y Fj(P)g Fi(\()p Fj(\033)1877 1226 y Fm(5)1938 1211 y Fi(+)2046 1171 y Fm(3)p Fl(\031)r(i)p 2046 1188 103 4 v 2080 1245 a Fm(2)2158 1211 y Fi(\))3888 1138 y Fy(\(121\))88 1428 y(Let)25 b(us)f(note)h(that)g(the)g(function)f Fj(f)1283 1443 y Fm(1)1322 1428 y Fi(\()p Fj(\033)1415 1443 y Fm(1)1455 1428 y Fo(j)p Fj(\033)1538 1443 y Fm(3)1578 1428 y Fo(j)p Fj(\033)1661 1443 y Fm(4)1700 1428 y Fj(;)17 b(\033)1799 1443 y Fm(5)1839 1428 y Fi(\))25 b Fy(is)g(anti-symmetric)e(w)-6 b(.r)h(.t.)30 b(the)25 b(transposition)e(of)i(the)g(last)g(tw)o(o)88 1548 y(v)n(ariables)30 b Fj(\033)531 1563 y Fm(4)609 1548 y Fo($)38 b Fj(\033)802 1563 y Fm(5)873 1548 y Fy(while)30 b(the)h(function)f Fj(f)1689 1563 y Fm(2)1728 1548 y Fi(\()p Fj(\033)1821 1563 y Fm(1)1861 1548 y Fj(;)17 b(\033)1960 1563 y Fm(3)2000 1548 y Fo(j)p Fj(\033)2083 1563 y Fm(5)2122 1548 y Fi(\))31 b Fy(is)f(symmetric)g(w)-6 b(.r)h(.t.)47 b(the)30 b(transposition)f(of)i(the)f(\002rst)88 1668 y(tw)o(o)24 b(v)n(ariables)g Fj(\033)698 1683 y Fm(1)766 1668 y Fo($)j Fj(\033)948 1683 y Fm(3)988 1668 y Fy(.)234 1789 y(Let)k(us)h(also)f(note)g(that)h(the)f(function)g Fj(f)1661 1804 y Fm(1)1732 1789 y Fy(de\002ned)i(by)e(\(118\))h(has)f (a)h(similar)f(to)g(the)h(case)g Fj(n)40 b Fi(=)h(5)31 b Fy(property)-6 b(,)88 1909 y(namely)g(,)23 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Fo(j)p Fj(\033)1671 2289 y Fm(4)1711 2274 y Fj(;)17 b(\033)1810 2289 y Fm(5)1849 2274 y Fi(\))28 b(=)f Fj(\037)2079 2289 y Fm(1)2119 2274 y Fi(\()p Fj(\033)2212 2289 y Fm(1)2252 2274 y Fo(j)p Fj(\033)2335 2289 y Fm(3)2374 2274 y Fo(j)p Fj(\033)2457 2289 y Fm(5)2497 2274 y Fi(\))765 2420 y Fy(Res)914 2435 y Fl(\033)954 2444 y Fg(4)989 2435 y Fm(=)p Fl(\033)1084 2444 y Fg(3)1119 2435 y Fk(\000)p Fl(\031)r(i)1245 2420 y Fj(f)1293 2435 y Fm(1)1333 2420 y Fi(\()p Fj(\033)1426 2435 y Fm(1)1465 2420 y Fo(j)p Fj(\033)1548 2435 y Fm(3)1588 2420 y Fo(j)p Fj(\033)1671 2435 y Fm(4)1711 2420 y Fj(;)17 b(\033)1810 2435 y Fm(5)1849 2420 y Fi(\))28 b(=)f Fj(\037)2079 2435 y Fm(3)2119 2420 y Fi(\()p Fj(\033)2212 2435 y Fm(1)2252 2420 y Fo(j)p Fj(\033)2335 2435 y Fm(3)2396 2420 y Fo(\000)c Fj(\031)t(i)p Fo(j)p Fj(\033)2671 2435 y Fm(5)2711 2420 y Fi(\))1139 b Fy(\(122\))88 2640 y(where)25 b(the)g(functions)f Fj(\037)960 2655 y Fm(1)1024 2640 y Fy(and)h Fj(\037)1254 2655 y Fm(3)1319 2640 y 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Fj(\033)1867 3290 y Fm(2)1906 3275 y Fo(j)p Fj(\033)1989 3290 y Fm(1)2029 3275 y Fj(;)17 b(\033)2128 3290 y Fm(3)2167 3275 y Fi(\))2205 3134 y Fn(\023)2306 3275 y Fi(=)28 b Fj(\016)2453 3290 y Fl(\033)2493 3299 y Fg(1)2532 3134 y Fn(\022)2705 3207 y Fj(\037)2766 3222 y Fm(1)2806 3207 y Fi(\()p Fj(\033)2899 3222 y Fm(4)2938 3207 y Fo(j)p Fj(\033)3021 3222 y Fm(1)3083 3207 y Fo(\000)23 b Fj(\031)t(i)p Fo(j)p Fj(\033)3358 3222 y Fm(3)3398 3207 y Fi(\))p 2615 3252 910 4 v 2615 3347 a Fj(P)14 b Fi(\()p Fj(\033)2785 3362 y Fm(1)2847 3347 y Fo(\000)2956 3308 y Fl(\031)r(i)p 2956 3324 67 4 v 2972 3381 a Fm(2)3033 3347 y Fi(\))p Fj(P)g Fi(\()p Fj(\033)3241 3362 y Fm(1)3302 3347 y Fi(+)3410 3308 y Fl(\031)r(i)p 3410 3324 V 3426 3381 a Fm(2)3487 3347 y Fi(\))394 3559 y(+)609 3492 y Fj(\037)670 3507 y Fm(3)710 3492 y Fi(\()p Fj(\033)803 3507 y Fm(4)843 3492 y Fo(j)p Fj(\033)926 3507 y Fm(1)987 3492 y Fo(\000)23 b Fj(\031)t(i)p Fo(j)p Fj(\033)1262 3507 y Fm(3)1302 3492 y Fi(\))p 502 3536 946 4 v 502 3631 a Fj(P)14 b Fi(\()p Fj(\033)672 3646 y Fm(1)733 3631 y Fo(\000)843 3592 y Fl(\031)r(i)p 843 3608 67 4 v 859 3666 a Fm(2)920 3631 y Fi(\))p Fj(P)g Fi(\()p Fj(\033)1128 3646 y Fm(1)1189 3631 y Fi(+)1297 3592 y Fm(3)p Fl(\031)r(i)p 1297 3608 103 4 v 1330 3666 a Fm(2)1409 3631 y Fi(\))1479 3559 y(+)1670 3492 y Fj(\037)1731 3507 y Fm(3)1771 3492 y Fi(\()p Fj(\033)1864 3507 y Fm(4)1903 3492 y Fo(j)p Fj(\033)1986 3507 y Fm(1)2048 3492 y Fo(\000)23 b Fi(2)p Fj(\031)t(i)p Fo(j)p Fj(\033)2372 3507 y Fm(3)2411 3492 y Fi(\))p 1587 3536 946 4 v 1587 3631 a Fj(P)14 b Fi(\()p Fj(\033)1757 3646 y Fm(1)1818 3631 y Fo(\000)1928 3592 y Fm(3)p Fl(\031)r(i)p 1928 3608 103 4 v 1961 3666 a Fm(2)2040 3631 y Fi(\))p Fj(P)g Fi(\()p Fj(\033)2248 3646 y Fm(1)2310 3631 y Fi(+)2417 3592 y Fl(\031)r(i)p 2417 3608 67 4 v 2433 3666 a Fm(2)2494 3631 y Fi(\))2542 3419 y Fn(\023)3888 3559 y Fy(\(123\))234 3841 y(In)30 b(order)g(to)g(check)h(the)f(sk)o(e)n(w-symmetry)e(of)i (the)g(answer)g(\(110\))g(one)g(can)h(follo)n(w)d(the)i(same)g(w)o(ay)h (lik)o(e)e(for)88 3962 y(the)h(pre)n(vious)f(case)i Fj(n)39 b Fi(=)f(5)p Fy(.)47 b(All)30 b(necessary)h(relations)f(may)g(be)h (straightforw)o(ardly)e(generalized)i(for)g(the)f(case)88 4082 y Fj(n)d Fi(=)h(6)d Fy(also.)30 b(F)o(or)25 b(instance,)f(the)h (inte)o(gral)e(formula)i(\(104\))f(may)h(be)g(generalized)g(as)g(follo) n(ws)395 4213 y Fn(Z)451 4438 y Fm(\000)495 4447 y Fg(3)560 4281 y Fj(d\033)666 4296 y Fm(3)p 560 4325 146 4 v 562 4417 a Fi(2)p Fj(\031)t(i)715 4348 y(X)796 4363 y Fm(3)836 4348 y Fi(\()p Fj(\033)929 4363 y Fm(3)969 4348 y Fi(\)\010\()p Fj(\033)1170 4363 y Fm(3)1210 4348 y Fi(\))p Fj(D)1329 4363 y Fm(3)1368 4348 y Fj(f)1416 4363 y Fm(1)1456 4348 y Fi(\()p Fj(\033)1549 4363 y Fm(4)1589 4348 y Fo(j)p Fj(\033)1672 4363 y Fm(3)1711 4348 y Fo(j)p Fj(\033)1794 4363 y Fm(1)1834 4348 y Fj(;)17 b(\033)1933 4363 y Fm(2)1972 4348 y Fi(\))406 4625 y(=)28 b Fj(D)591 4640 y Fm(1)630 4485 y Fn(\022)704 4625 y Fj(p)753 4640 y Fm(3)792 4625 y Fi(\()p Fj(\033)885 4640 y Fm(1)925 4625 y Fi(\))973 4558 y Fj(\037)1034 4573 y Fm(3)1073 4558 y Fi(\()p Fj(\033)1166 4573 y Fm(4)1206 4558 y Fo(j)p Fj(\033)1289 4573 y Fm(1)1351 4558 y Fo(\000)22 b Fj(\031)t(i)p Fo(j)p Fj(\033)1625 4573 y Fm(2)1665 4558 y Fi(\))p 973 4602 731 4 v 1110 4697 a Fj(P)14 b Fi(\()p Fj(\033)1280 4712 y Fm(1)1341 4697 y Fo(\000)1451 4658 y Fl(\031)r(i)p 1451 4674 67 4 v 1467 4732 a Fm(2)1528 4697 y Fi(\))1735 4625 y Fo(\000)23 b Fj(p)1884 4640 y Fm(3)1923 4625 y Fi(\()p Fj(\033)2016 4640 y Fm(1)2078 4625 y Fi(+)f(2)p Fj(\031)t(i)p Fi(\))2365 4558 y Fj(\037)2426 4573 y Fm(3)2466 4558 y Fi(\()p Fj(\033)2559 4573 y Fm(4)2598 4558 y Fo(j)p Fj(\033)2681 4573 y Fm(1)2721 4558 y Fo(j)p Fj(\033)2804 4573 y Fm(2)2844 4558 y Fi(\))p 2365 4602 517 4 v 2378 4697 a Fj(P)14 b Fi(\()p Fj(\033)2548 4712 y Fm(1)2610 4697 y Fi(+)2718 4658 y Fm(5)p Fl(\031)r(i)p 2718 4674 103 4 v 2751 4732 a Fm(2)2830 4697 y Fi(\))2891 4485 y Fn(\023)401 4909 y Fo(\000)22 b Fj(D)581 4924 y Fm(2)621 4769 y Fn(\022)694 4909 y Fj(p)743 4924 y Fm(3)783 4909 y Fi(\()p Fj(\033)876 4924 y Fm(2)915 4909 y Fi(\))963 4842 y Fj(\037)1024 4857 y Fm(3)1064 4842 y Fi(\()p Fj(\033)1157 4857 y Fm(4)1197 4842 y Fo(j)p Fj(\033)1280 4857 y Fm(2)1341 4842 y Fo(\000)h Fj(\031)t(i)p Fo(j)p Fj(\033)1616 4857 y Fm(1)1656 4842 y Fi(\))p 963 4886 731 4 v 1101 4981 a Fj(P)14 b Fi(\()p Fj(\033)1271 4996 y Fm(2)1332 4981 y Fo(\000)1442 4942 y Fl(\031)r(i)p 1442 4959 67 4 v 1457 5016 a Fm(2)1518 4981 y Fi(\))1726 4909 y Fo(\000)22 b Fj(p)1874 4924 y Fm(3)1914 4909 y Fi(\()p Fj(\033)2007 4924 y Fm(2)2069 4909 y Fi(+)g(2)p Fj(\031)t(i)p Fi(\))2356 4842 y Fj(\037)2417 4857 y Fm(3)2456 4842 y Fi(\()p Fj(\033)2549 4857 y Fm(4)2589 4842 y Fo(j)p Fj(\033)2672 4857 y Fm(2)2712 4842 y Fo(j)p Fj(\033)2795 4857 y Fm(1)2834 4842 y Fi(\))p 2356 4886 517 4 v 2369 4981 a Fj(P)14 b Fi(\()p Fj(\033)2539 4996 y Fm(2)2600 4981 y Fi(+)2708 4942 y Fm(5)p Fl(\031)r(i)p 2708 4959 103 4 v 2742 5016 a Fm(2)2821 4981 y Fi(\))2882 4769 y Fn(\023)401 5193 y Fo(\000)22 b Fj(p)549 5208 y Fm(3)589 5193 y Fi(\()p Fj(\033)682 5208 y Fm(1)722 5193 y Fi(\))p Fj(\016)807 5152 y Fk(\000)p Fm(1)803 5218 y Fl(\033)843 5227 y Fg(1)901 5053 y Fn(\024)963 5126 y Fy(Res)1113 5141 y Fl(\033)r Fm(3=)p Fl(\033)r Fm(1+)p Fl(\031)r(i)p 963 5171 487 4 v 979 5266 a Fj(P)14 b Fi(\()p Fj(\033)1149 5281 y Fm(1)1211 5266 y Fi(+)1319 5226 y Fl(\031)r(i)p 1319 5243 67 4 v 1335 5300 a Fm(2)1396 5266 y Fi(\))1460 5053 y Fn(\022)1533 5193 y Fj(D)1614 5208 y Fm(1)1653 5193 y Fj(f)1701 5208 y Fm(1)1741 5193 y Fi(\()p Fj(\033)1834 5208 y Fm(4)1874 5193 y Fo(j)p Fj(\033)1957 5208 y Fm(1)1996 5193 y Fo(j)p Fj(\033)2079 5208 y Fm(3)2119 5193 y Fj(;)j(\033)2218 5208 y Fm(2)2257 5193 y Fi(\))22 b Fo(\000)h Fj(D)2498 5208 y Fm(3)2538 5193 y Fj(f)2586 5208 y Fm(1)2625 5193 y Fi(\()p Fj(\033)2718 5208 y Fm(4)2758 5193 y Fo(j)p Fj(\033)2841 5208 y Fm(3)2880 5193 y Fo(j)p Fj(\033)2963 5208 y Fm(1)3003 5193 y Fj(;)17 b(\033)3102 5208 y Fm(2)3142 5193 y Fi(\))3180 5053 y Fn(\023\025)401 5478 y Fo(\000)22 b Fj(p)549 5493 y Fm(3)589 5478 y Fi(\()p Fj(\033)682 5493 y Fm(2)722 5478 y Fi(\))p Fj(\016)807 5437 y Fk(\000)p Fm(1)803 5502 y Fl(\033)843 5511 y Fg(2)901 5337 y Fn(\024)963 5410 y Fy(Res)1113 5425 y Fl(\033)r Fm(3=)p Fl(\033)r Fm(2+)p Fl(\031)r(i)p 963 5455 487 4 v 979 5550 a Fj(P)14 b Fi(\()p Fj(\033)1149 5565 y Fm(2)1211 5550 y Fi(+)1319 5511 y Fl(\031)r(i)p 1319 5527 67 4 v 1335 5584 a Fm(2)1396 5550 y Fi(\))1460 5337 y Fn(\022)1533 5478 y Fj(D)1614 5493 y Fm(2)1653 5478 y Fj(f)1701 5493 y Fm(1)1741 5478 y Fi(\()p Fj(\033)1834 5493 y Fm(4)1874 5478 y Fo(j)p Fj(\033)1957 5493 y Fm(2)1996 5478 y Fo(j)p Fj(\033)2079 5493 y Fm(1)2119 5478 y Fj(;)j(\033)2218 5493 y Fm(3)2257 5478 y Fi(\))22 b Fo(\000)h Fj(D)2498 5493 y Fm(3)2538 5478 y Fj(f)2586 5493 y Fm(1)2625 5478 y Fi(\()p Fj(\033)2718 5493 y Fm(4)2758 5478 y Fo(j)p Fj(\033)2841 5493 y Fm(3)2880 5478 y Fo(j)p Fj(\033)2963 5493 y Fm(1)3003 5478 y Fj(;)17 b(\033)3102 5493 y Fm(2)3142 5478 y Fi(\))3180 5337 y Fn(\023\025)3888 5478 y Fy(\(124\))2046 5871 y(31)p eop %%Page: 32 32 32 31 bop 88 289 a Fy(Only)22 b(one)h(complication)e(in)i(comparison)e (with)i Fj(n)28 b Fi(=)f(5)c Fy(appears)g(because)h(of)f(the)f(second)h (term)g(with)f Fj(D)3773 304 y Fm(2)3812 289 y Fi(\()p Fj(:)17 b(:)g(:)p Fi(\))23 b Fy(in)88 409 y(the)g(r)-5 b(.h.s.)29 b(of)23 b(\(124\))g(that)g(mak)o(es)g(a)g(non-zero)g (contrib)n(ution)f(after)h(further)h(inte)o(gration.)k(Because)c(of)f (this)g(f)o(act)g(we)88 529 y(had)28 b(to)g(include)g(an)g(additional)f (terms)h(with)g(the)g(ratio)g Fj(P)14 b Fi(\()p Fj(\033)2224 544 y Fm(3)2288 529 y Fi(+)2399 490 y Fm(3)p Fl(\031)r(i)p 2399 507 103 4 v 2433 564 a Fm(2)2511 529 y Fi(\))p Fj(=P)g Fi(\()p Fj(\033)2768 544 y Fm(3)2832 529 y Fi(+)2943 490 y Fm(7)p Fl(\031)r(i)p 2943 507 V 2976 564 a Fm(2)3055 529 y Fi(\))28 b Fy(in)g(\(120\))h(which)f(appeared)88 650 y(to)c(be)h(necessary)g(for)g(the)g(correct)g(de\002nition)f(of)h (the)g(function)f Fj(f)2377 665 y Fm(1)2416 650 y Fy(.)88 987 y Fp(Refer)m(ences)541 1210 y Fy([1])49 b(F)-8 b(.A.Smirno)o(v)i(,) 28 b Fw(F)-10 b(orm)27 b(F)-7 b(actor)o(s)27 b(in)h(Completely)g(Inte)l (gr)o(able)f(Models)g(of)h(Quantum)g(F)l(ield)f(The-)706 1331 y(ory)-5 b(.)24 b Fy(Adv)-6 b(.)24 b(Series)i(in)e(Math.)g(Phys.)g (14,)h(W)-8 b(orld)24 b(Scienti\002c,)h(Singapore)g(\(1992\))541 1529 y([2])49 b(I.)25 b(Frenk)o(el,)g(N.)g(Reshetikhin,)f(Comm.)g (Math.)g(Phys.)g(\(1992\))h(1-60)541 1728 y([3])49 b(M.)25 b(Jimbo,)e(T)-7 b(.)24 b(Miw)o(a,)h(J.Phys.)e(A29)i(\(1996\))f (2923-2958)541 1927 y([4])49 b(F)-8 b(.A.)25 b(Smirno)o(v)-6 b(,)23 b(Int.J.Mod.Phys.A7,)f(\(1992\))j(S813-858)541 2126 y([5])49 b(F)-8 b(.A.Smirno)o(v)i(,)23 b(J.Phys.A)h(19)h(\(1986\)) f(L575-L578,)541 2324 y([6])49 b(F)-8 b(.A.)25 b(Smirno)o(v)-6 b(,)23 b(Lett.Math.Phys.)g(36)i(\(1996\))f(267)541 2523 y([7])49 b(F)-8 b(.A.Smirno)o(v)i(,)23 b(Nucl.)i(Phys.)f(B453[FS])i (\(1995\))e(807)541 2722 y([8])49 b(J.M.)24 b(Maillet,)g(V)-13 b(.)25 b(T)-7 b(erras,)25 b Fw(Nucl.)g(Phys.)g Fa(B575)f Fy(\(2000\))h(627-647.)541 2920 y([9])49 b(N.)25 b(Kitanine,)f(J.)g(M.) h(Maillet,)e(V)-13 b(.)25 b(T)-7 b(erras,)26 b Fw(Nucl.)f(Phys.)f Fa(B)h(567)g Fy(\(2000\),)f(no.)h(3,)f(554\226582.)491 3119 y([10])49 b(H.E.)25 b(Boos,)f(V)-13 b(.E.)25 b(K)m(orepin,)f Fw(J)n(.)g(Phys.)h(A:)g(Math.)f(Gen)h Fy(34)f(\(2001\))h(5311-5316)491 3318 y([11])49 b(H.E.)25 b(Boos,)f(V)-13 b(.E.)25 b(K)m(orepin,)706 3438 y(\223Ev)n(aluation)36 b(of)i(inte)o(grals)f(representing)g (correlations)g(in)g(XXX)h(Heisenber)n(g)g(spin)e(chain\224,)706 3559 y Fw(Pr)l(o)o(gr)l(ess)21 b(in)h(Mathematical)f(Physics)h Fy(V)-13 b(ol.)22 b Fa(23)p Fy(,)g(MathPhys)f(Odysse)o(y)g(2001,)h Fw(\224Inte)l(gr)o(able)f(Sys-)706 3679 y(tems)40 b(and)g(Be)m (yond\224)h Fy(,)j(in)c(honor)g(of)g(Barry)h(M.)81 b(McCo)o(y)-6 b(,)43 b(edited)d(by)g(M.)g(Kashiw)o(ara)g(and)706 3799 y(T)-7 b(.)25 b(Miw)o(a)f(\(Birkh)6 b(\250)-39 b(auser)l(,)25 b(Boston,)f(2002\))g(65-108;)g(hep-th/0105144.)491 3998 y([12])49 b(H.E.)28 b(Boos,)h(V)-13 b(.E.)28 b(K)m(orepin,)g(F)-8 b(.A.)28 b(Smirno)o(v)f(,)i Fw(Nucl.)f(Phys.)g(B)g Fy(V)-13 b(ol.)27 b(658/3)g(\(2003\))h(417)g(-)g(439;)706 4119 y(hep-th/0209246.)491 4317 y([13])49 b(H.E.)44 b(Boos,)50 b(V)-13 b(.E.)44 b(K)m(orepin,)k(F)-8 b(.A.)45 b(Smirno)o(v)-6 b(,)48 b Fw(\223Ne)o(w)d(F)-10 b(ormulae)43 b(for)h(Solutions)f(of)h (Quan-)706 4438 y(tum)29 b(Knizhnik-Zamolodc)o(hik)o(o)o(v)f(Equations) f(on)i(Le)o(vel)h(-4\224)p Fy(,)h(hep-th/0304077;)e(submitted)f(to)706 4558 y Fw(J)n(.Phys.)c Fy(A.)491 4757 y([14])49 b(H.E.)32 b(Boos,)j(V)-13 b(.E.)32 b(K)m(orepin,)h(Y)-13 b(.)33 b(Nishiyama)e(and)i(M.)f(Shiroishi,)h Fw(J)n(.)f(Phys.A:)g(Math.)g(Gen) h Fa(35)706 4877 y Fy(\(2002\))25 b(4443;)f(cond-mat/0202346.)491 5076 y([15])49 b(A.)25 b(Nakayashiki,)f(F)-8 b(.A.)25 b(Smirno)o(v)-6 b(,)23 b(Comm.)h(Math.)g(Phys.,)g(217)g(\(2001\),)h (623)491 5275 y([16])49 b(A.)30 b(Nakayashiki,)g Fw(On)g(the)g (cohomolo)o(gy)f(of)g(theta)g(divisor)g(of)h(hyper)l(elliptic)e(J)m (acobian.)h Fy(Con-)706 5395 y(temporary)50 b(mathemathics,)k(v)n(ol)49 b(309,)56 b(in)49 b(Inte)o(grable)h(systems,)k(topology)48 b(and)i(physics,)706 5515 y(M.Guest)24 b(et)h(al.)g(ed.,)f(AMS)h (\(2002\))2046 5871 y(32)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0305171006305--