New results on the Bochner condition about classical orthogonal polynomials

Loureiro, Ana F. (2010) New results on the Bochner condition about classical orthogonal polynomials. Journal of Mathematical Analysis and Applications, 364 (2). pp. 307-323. ISSN 0022-247X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)

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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 Fk with polynomial coefficients defined by a recursive relation. Here, an explicit expression of Fk for any positive integer k is given. The main aim of this work is to explicitly establish sums relating any power of F with Fk, 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.

Item Type: Article
Uncontrolled keywords: Classical orthogonal polynomials; Bochner differential equation; Stirling numbers; Bessel–Stirling numbers; Jacobi–Stirling numbers; Inverse relations
Subjects: Q Science > QA Mathematics (inc Computing science) > QA351 Special functions
Divisions: Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science > Applied Mathematics
Depositing User: Ana F. Loureiro
Date Deposited: 11 Oct 2012 15:17
Last Modified: 12 Feb 2013 16:52
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