[Paper Review] Combinatorics of $(q,y)$-Laguerre polynomials and their moments

Qiongqiong Pan, Jiang Zeng|arXiv (Cornell University)|Jan 3, 2019

Advanced Combinatorial Mathematics16 references4 citations

TL;DR

This paper introduces a combinatorial model for $(q,y)$-Laguerre polynomials using colored Foata-Strehl Laguerre configurations, establishing a bijection with rook placements and matching polynomials. The key contribution is proving that linearization coefficients are polynomials in $y$ and $q$ with nonnegative integer coefficients, extending classical Laguerre combinatorics to a two-parameter $q$-analogue via $q$-Riordan matrix theory and continued fractions.

ABSTRACT

We consider a $(q,y)$-analogue of Laguerre polynomials $L^{(α)}_n(x;y;q)$ for integral $α\geq -1$, which turns out to be a rescaled version of Al-Salam--Chihara polynomials. A combinatorial interpretation for the $(q,y)$-Laguerre polynomials is given using a colored version of Foata-Strehl's Laguerre configurations with suitable statistics. When $α\geq 0$, the corresponding moments are described using certain classical statistics on permutations, and the linearization coefficients are proved to be a polynomial in $y$ and $q$ with nonnegative integral coefficients.

Motivation & Objective

• To develop a unified combinatorial model for $(q,y)$-Laguerre polynomials that generalizes both Al-Salam–Chihara and $q$-Laguerre polynomials.
• To interpret the moments of $(q,y)$-Laguerre polynomials combinatorially using classical permutation statistics.
• To prove that the linearization coefficients of these polynomials are polynomials in $y$ and $q$ with nonnegative integer coefficients.
• To establish a bijection between $(q,y)$-Laguerre configurations and rook placements or matching polynomials on complete bipartite graphs.

Proposed method

• Introduce a colored version of Foata and Strehl’s Laguerre configurations, assigning colors to cycles and paths based on $q$-statistics.
• Define key statistics: inversion number $\mathsf{inv}$, cycle weight $\mathsf{cw}$, cycle degree $\mathsf{cd}$, and index $\mathsf{ind}$, to encode $q$ and $y$ weights.
• Construct a bijection $\phi$ from $(q,y)$-Laguerre configurations to rook placements on a Ferrers board, preserving statistics and enabling generating function derivation.
• Use the theory of $q$-Riordan matrices and continued fractions to analyze moment sequences and linearization coefficients.
• Establish a connection between $(q,y)$-Laguerre polynomials and matching polynomials of complete bipartite graphs $K_{n,n+\alpha}$, generalizing the classical identity $m(K_{n,n+\alpha},x) = x^\alpha L_n^{(\alpha)}(x^2)$.
• Prove that the linearization coefficients are in $\mathbb{Z}_{\geq 0}[y,q]$ by showing their combinatorial interpretation via weighted configurations.

Experimental results

Research questions

• RQ1 How can a combinatorial model for $(q,y)$-Laguerre polynomials be constructed that unifies known special cases like Al-Salam–Chihara and $q$-Laguerre polynomials?
• RQ2 What statistics on colored Laguerre configurations generate the $(q,y)$-Laguerre polynomials via generating functions?
• RQ3 Are the linearization coefficients of $(q,y)$-Laguerre polynomials polynomials in $y$ and $q$ with nonnegative integer coefficients?
• RQ4 Can the moments of $(q,y)$-Laguerre polynomials be interpreted combinatorially using permutation statistics such as inversions and cycle counts?
• RQ5 Is there a natural bijection between $(q,y)$-Laguerre configurations and rook placements or matchings in complete bipartite graphs?

Key findings

• The $(q,y)$-Laguerre polynomials $L_n^{(\alpha)}(x;y|q)$ are shown to be a rescaled version of Al-Salam–Chihara polynomials, extending the classical Laguerre case.
• A colored Laguerre configuration model is constructed where $y$ tracks cycle count and $q$ tracks inversion and cycle degree statistics.
• The linearization coefficients of $L_n^{(\alpha)}(x;y|q)$ are proven to be in $\mathbb{Z}_{\geq 0}[y,q]$, with explicit combinatorial interpretations via weighted configurations.
• A bijection $\phi$ is established between $(q,y)$-Laguerre configurations and rook placements on a Ferrers board, preserving key statistics: $\mathsf{inv}(C) = \mathsf{inv}(\rho)$, $\mathsf{cw}(B) + \mathsf{ind}(\mathcal{R}) = |\underline{\sigma}| + \mathsf{rl}(\lambda)$.
• The moments of $L_n^{(\alpha)}(x;y|q)$ for $\alpha \in \mathbb{N}_0$ are interpreted combinatorially using classical permutation statistics, and the generating function is derived via $q$-Riordan matrix techniques.
• A matching polynomial model is constructed for $L_n^{(\alpha)}(x;y|q)$, generalizing the identity $m(K_{n,n+\alpha},x) = x^\alpha L_n^{(\alpha)}(x^2)$ to the $(q,y)$-case via a bijection to $\mathcal{LC}_{n,k}^{(\alpha)}$.