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 SIGMA, 2018, Volume 14, 063, 27 pages (Mi sigma1362)

$d$-Orthogonal Analogs of Classical Orthogonal Polynomials

Emil Horozovab

a Institute of Mathematics and Informatics, Bulg. Acad. of Sci., Acad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria
b Department of Mathematics and Informatics, Sofia University, 5 J. Bourchier Blvd., Sofia 1126, Bulgaria

Abstract: Classical orthogonal polynomial systems of Jacobi, Hermite and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner they are the only systems on the real line with this property. Similar results hold for the discrete orthogonal polynomials. In a recent paper we introduced a natural class of polynomial systems whose members are the eigenfunctions of a differential operator of higher order and which are orthogonal with respect to $d$ measures, rather than one. These polynomial systems, enjoy a number of properties which make them a natural analog of the classical orthogonal polynomials. In the present paper we continue their study. The most important new properties are their hypergeometric representations which allow us to derive their generating functions and in some cases also Mehler–Heine type formulas.

Keywords: $d$-orthogonal polynomials; finite recurrence relations; bispectral problem; generalized hypergeometric functions; generating functions.

 Funding Agency Grant Number Bulgarian Fund “Scientific research” DN 02-5 This research has been partially supported by the Grant No DN 02-5 of the Bulgarian Fund “Scientific research”.

DOI: https://doi.org/10.3842/SIGMA.2018.063

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MSC: 34L20; 30C15; 33E05
Received: October 1, 2017; in final form June 13, 2018; Published online June 26, 2018
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Citation: Emil Horozov, “$d$-Orthogonal Analogs of Classical Orthogonal Polynomials”, SIGMA, 14 (2018), 063, 27 pp.

Citation in format AMSBIB
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