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Constructing Bispectral Orthogonal Polynomials from the Classical Discrete Families of Charlier, Meixner and Krawtchouk

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Abstract

Given a sequence of polynomials \((p_n)_n\), an algebra of operators \({\mathcal A}\) acting in the linear space of polynomials, and an operator \(D_p\in {\mathcal A}\) with \(D_p(p_n)=np_n\), we form a new sequence of polynomials \((q_n)_n\) by considering a linear combination of \(m+1\) consecutive \(p_n\): \(q_n=p_n+\sum _{j=1}^m\beta _{n,j}p_{n-j}\). Using the concept of \(\mathcal {D}\)-operator, we determine the structure of the sequences \(\beta _{n,j}, j=1,\ldots ,m,\) so that the polynomials \((q_n)_n\) are eigenfunctions of an operator in the algebra \({\mathcal A}\). As an application, from the classical discrete families of Charlier, Meixner, and Krawtchouk, we construct orthogonal polynomials \((q_n)_n\) which are also eigenfunctions of higher-order difference operators.

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de Sevilla, APDO, P. O. Box 1160, 41080 , Seville, Spain

    Antonio J. Durán & Manuel D. de la Iglesia

Authors
  1. Antonio J. Durán
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  2. Manuel D. de la Iglesia
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Corresponding author

Correspondence to Manuel D. de la Iglesia.

Additional information

Communicated by Tom H. Koornwinder.

Partially supported by MTM2012-36732-C03-03 (Ministerio de Economía y Competitividad), FQM-262, FQM-4643, FQM-7276 (Junta de Andalucía) and Feder Funds (European Union).

The authors would like to thank an anonymous referee for his/her comments and suggestions.

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Durán, A.J., de la Iglesia, M.D. Constructing Bispectral Orthogonal Polynomials from the Classical Discrete Families of Charlier, Meixner and Krawtchouk. Constr Approx 41, 49–91 (2015). https://doi.org/10.1007/s00365-014-9251-5

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  • DOI: https://doi.org/10.1007/s00365-014-9251-5

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