 2089 Frederick Ira Moxley III
 Periodic Solutions of Inverse Quantum Orthogonal Equations
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Sep 26, 20

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Abstract. In the year $1939$, the Mathematician G.H. Hardy proved that the only functions $f$ which satisfy the classical orthogonality relation
%
egin{align}
onumber \int_{0}^{1}f(\lambda_{m}t)f(\lambda_{n}t)dt =0, \quad m
eq n,
nd{align}
%
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 RiemannStieltjes 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}
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