New results on the Bochner condition about classical orthogonal polynomials |
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Authors: | Ana F. Loureiro |
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Affiliation: | CMUP & Departamento de Física e Matemática, Instituto Superior de Engenharia de Coimbra, Rua Pedro Nunes, Quinta da Nora, 3030-199 Coimbra, Portugal |
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Abstract: | 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. |
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Keywords: | Classical orthogonal polynomials Bochner differential equation Stirling numbers Bessel-Stirling numbers Jacobi-Stirling numbers Inverse relations |
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