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

https://doi.org/10.1016/j.jmaa.2022.126358Get rights and content
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Abstract

The purpose of this note is to characterize those orthogonal polynomials sequences (Pn)n0 for whichπ(x)DqPn(x)=(anx+bn)Pn(x)+cnPn1(x),n=0,1,2,, where Dq is the Askey-Wilson operator, π is a polynomial of degree at most 2, and (an)n0, (bn)n0 and (cn)n0 are sequences of complex numbers such that cn0 for n=1,2,.

Keywords

Askey-Wilson operator
Al-Salam-Chihara polynomials
Chebyshev polynomials of the first kind
Continuous q-Jacobi polynomials

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Our beloved friend and mentor J. Petronilho passed away unexpectedly on August 27, 2021 while we were preparing the final version of this manuscript. To him, our eternal gratitude.