Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials

Authors:
Mourad E. H. Ismail;Anisse Kasraoui;Jiang Zeng
Affiliations:
City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong and King Saud University, Riyadh, Saudi Arabia;Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Vienna, Austria;Université de Lyon, Université Lyon 1, Institut Camille Jordan, UMR 5028 du CNRS, 69622 Villeurbanne, France
Venue:
Journal of Combinatorial Theory Series A
Year:
2013

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Abstract

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.