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M. Ismail, A. Kasraoui and J. Zeng

# Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials

**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.

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