Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:120

Article

Ismail, Mourad E. H.^1,2  Kasraoui, Anisse³  Zeng, Jiang⁴ 

[1] City Univ Hong Kong, Kowloon, Hong Kong, Peoples R China

[2] King Saud Univ, Riyadh, Saudi Arabia

[3] Univ Vienna, Fak Math, A-1090 Vienna, Austria

[4] Univ Lyon 1, Inst Camille Jordan, UMR 5028, CNRS, F-69622 Villeurbanne, France

关键词: Orthogonal polynomials;    Separation of variables;    Linearization coefficients;    Sheffer polynomials;    q-Analogues;    Derangements;   

DOI  :  10.1016/j.jcta.2012.10.007

来源: Elsevier

【 摘 要 】

We propose a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials. We first establish a difference system and then solve it combinatorially and analytically using the method of separation of variables. We illustrate our approach by applying it to determine the number of perfect matchings, derangements, and other weighted permutation problems. The separation of variables technique naturally leads to integral representations of combinatorial numbers where the integrand contains a product of one or more types of orthogonal polynomials. This also establishes the positivity of such integrals. (c) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

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