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arXiv · 2109.06147

A characterization of continuous $q$-Jacobi, Chebyshev of the first kind and Al-Salam Chihara polynomials

K. Castillo·D. Mbouna·J. Petronilho
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Abstract

The purpose of this note is to characterize those orthogonal polynomials sequences $(P_n)_{n\geq0}$ for which $$ \pi(x)\mathcal{D}_q P_n(x)=(a_n x+b_n)P_n(x)+c_n P_{n-1}(x),\quad n=0,1,2,\dots, $$ where $\mathcal{D}_q$ is the Askey-Wilson operator, $\pi$ is a polynomial of degree at most 2, and $(a_n)_{n\geq0}$, $(b_n)_{n\geq0}$ and $(c_n)_{n\geq0}$ are sequences of complex numbers such that $c_n\neq0$ for $n=1,2,\dots$.

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K. Castillo, D. Mbouna, J. Petronilho. 2021-09-13. A characterization of continuous $q$-Jacobi, Chebyshev of the first kind and Al-Salam Chihara polynomials. https://arxiv.org/abs/2109.06147

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