Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials

TL;DR

This paper introduces a novel method combining separation of variables with combinatorial analysis to interpret linearization coefficients of orthogonal polynomials, linking them to permutation problems and integral representations.

Contribution

It develops a new approach to combinatorial interpretations of linearization coefficients using separation of variables, applicable to various orthogonal polynomials.

Findings

Derived integral representations involving orthogonal polynomials

Connected linearization coefficients to permutation enumeration

Established positivity of certain integral formulas

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.