Frederick Ira Moxley III
Periodic Solutions of Inverse Quantum Orthogonal Equations
(388K, .pdf)

ABSTRACT.  In the year $1939$, the Mathematician G.H. Hardy proved that the only functions $f$ which satisfy the classical orthogonality relation 
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egin{align} 

onumber \int_{0}^{1}f(\lambda_{m}t)f(\lambda_{n}t)dt =0, \quad m 
eq n, 
nd{align} 
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are the Bessel functions $J_{
u}(t)$ under certain constraints, where $
u>-1$ is the order of the Bessel function, and $\lambda_{m}$, $\lambda_{n}$ are the zeros of the Bessel function. More recently, the Mathematician L.D. Abreu proved that if a function $f\in\mathcal{L}_{q}^{2}(0,1)$ is $q$-orthogonal with respect to its own zeros in the interval $(0,1)$, then it satisfies the $q$-orthogonality relation 
% 
egin{align} 

onumber \int_{0}^{1}f(\lambda_{m}t)f(\lambda_{n}t)d_{q}t =0, \quad m 
eq n, 
nd{align} 
% 
where the $q$-integral is a Riemann-Stieltjes integral with respect to a step function having infinitely many points of increase at the points $q^{ll}$, with the step size at the point $q^{ll}$ being $q$, $orall\; ll \in \mathbb{N}_{0}$, where $\mathbb{N}_{0}:=\mathbb{N}