New results on the Bochner condition about classical orthogonal polynomials

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:364

Article

Loureiro, Ana F.^1,2 

[1] CMUP, P-3030199 Coimbra, Portugal

[2] Inst Super Engn Coimbra, Dept Fis & Matemat, P-3030199 Coimbra, Portugal

关键词: Classical orthogonal polynomials;    Bochner differential equation;    Stirling numbers;    Bessel-Stirling numbers;    Jacobi-Stirling numbers;    Inverse relations;   

DOI  :  10.1016/j.jmaa.2009.12.003

来源: Elsevier

【 摘 要 】

The classical polynomials (Hermite, Laguerre, Bessel and Jacobi) are the only orthogonal polynomial sequences (OPS) whose elements are eigenfunctions of the Bochner second-order differential operator f (Bochner, 1929 [3]). In Loureiro, Maroni and da Rocha (2006) [18] these polynomials were described as eigenfunctions of an even order differential operator f(k) with polynomial coefficients defined by a recursive relation. Here, an explicit expression of f(k) for any positive integer k is given. The main aim of this work is to explicitly establish sums relating any power of f with f(k), k >= 1, in other words, to bring a pair of inverse relations between these two operators. This goal is accomplished with the introduction of a new sequence of numbers: the so-called A-modified Stirling numbers, which could be also called as Bessel or Jacobi-Stirling numbers, depending on the context and the values of the complex parameter A. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

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