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Remarks on Askey–Wilson polynomials and Meixner polynomials of the second kind

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Abstract

The purpose of this note is to characterize all the sequences of orthogonal polynomials \((P_n)_{n\ge 0}\) such that

$$\begin{aligned} \frac{\triangle }{\mathbf{\triangle } x(s-1/2)}P_{n+1}(x(s-1/2))=c_n(\triangle +2\,\mathrm {I})P_n(x(s-1/2)), \end{aligned}$$

where \(\,\mathrm {I}\) is the identity operator, x defines a class of lattices with, generally, nonuniform step-size, and \(\triangle f(s)=f(s+1)-f(s)\). The proposed method can be applied to similar and to more general problems involving the mentioned operators, in order to obtain new characterization theorems for some specific families of classical orthogonal polynomials on lattices.

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Acknowledgements

The authors thank the referees for their useful comments.

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Authors and Affiliations

  1. CMUC, Dep. Mathematics, University of Coimbra, 3001-501, Coimbra, Portugal

    K. Castillo, D. Mbouna & J. Petronilho

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  1. K. Castillo
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  2. D. Mbouna
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  3. J. Petronilho
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Correspondence to D. Mbouna.

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This work is supported by the Centre for Mathematics of the University of Coimbra-UID/MAT/00324/2019, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.

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Castillo, K., Mbouna, D. & Petronilho, J. Remarks on Askey–Wilson polynomials and Meixner polynomials of the second kind. Ramanujan J 58, 1159–1170 (2022). https://doi.org/10.1007/s11139-021-00508-6

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  • DOI: https://doi.org/10.1007/s11139-021-00508-6

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