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Connection coefficients for ultraspherical polynomials with argument doubling and generalized bispectrality

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Abstract

We start by presenting a generalization of a discrete wave equation that is satisfied by the entries of the matrix coefficients of the refinement equation corresponding to the multiresolution analysis of Alpert. The entries are functions of two discrete variables and they can be expressed in terms of the Legendre polynomials. Next, we generalize these functions to the case of the ultraspherical polynomials and show that these new functions obey two generalized eigenvalue problems in each of the two discrete variables, which constitute a generalized bispectral problem.

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Acknowledgments

M. D. was supported in part by the NSF DMS grant 2008844. The authors are grateful to Erik Koelink for interesting and helpful remarks. They are also indebted to the anonymous referees for suggestions that helped to improve the presentation of the results.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, 341 Mansfield Road, U-1009, Storrs, CT, 06269-1009, USA

    Maxim Derevyagin

  2. School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA, 30332-0160, USA

    Jeffrey S. Geronimo

Authors
  1. Maxim Derevyagin
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  2. Jeffrey S. Geronimo
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Correspondence to Jeffrey S. Geronimo.

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Dedicated to the memory of Richard Askey

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Derevyagin, M., Geronimo, J.S. Connection coefficients for ultraspherical polynomials with argument doubling and generalized bispectrality. JAMA 150, 57–81 (2023). https://doi.org/10.1007/s11854-023-0271-6

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  • DOI: https://doi.org/10.1007/s11854-023-0271-6

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