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.2010 Mathematics Subject Classification. Primary 33D15, 05A15, Secondary 30E05, 33 C15.

On Linearization Coefficients of $q$-Laguerre Polynomials

“…For the remainder of this section we will find a combinatorial model for L n (x; q, y) in Theorem 2. 4 and give yet another expression for µ n (q, y) in (7). To do this, we define some statistics for matchings.…”

“…There are q-analogs of the above combinatorial formulas for linearization coefficients of Hermite, Charlier and Laguerre polynomials due to Ismail, Stanton and Viennot [8], Anshelevich [1] and Kasraoui, Stanton and Zeng [9], respectively. There is a unified way to prove combinatorial formulas for linearization coefficients using so called "separation of variables" [7]. We refer the reader to the survey [2] for more details on these linearization coefficients.…”

On moments of classical orthogonal polynomials

“…The generalized Charlier moments with respect to the basis x n have the representation µ n x k = e a a n . Note that (5.73) appears in [44] and [26] without the constant µ 0 = e a .…”

Application of the Orthogonal Polynomial Fitting Method in Estimating PM2.5 Concentrations in Central and Southern Regions of China

“…Orthogonal polynomial fitting is one of the methods combining accuracy with efficiency. The application of orthogonal polynomial fitting has been studied for decades [24,25,26,27,28,29,30]. In our study, an orthogonal polynomial fitting method based on Chebyshev basis functions is introduced to estimate PM 2.5 concentrations in central and southern regions of China.…”