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.
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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 ). In Loureiro, Maroni and da Rocha (2006)  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.
|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|
|Resource URI:||http://kar.kent.ac.uk/id/eprint/31561 (The current URI for this page, for reference purposes)|
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