Combinatorial formulas for the coefficients of the Al-Salam-Chihara polynomials

The Combinatorics of Al-Salam-Chihara $q$-Laguerre polynomials

We describe various aspects of the Al-Salam-Chihara $q$-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients. It is remarkable that the corresponding moment sequence appears also in the recent work of Postnikov and Williams on enumeration of totally positive Grassmann cells.

The Andrews-Stanley partition function and Al-Salam-Chihara polynomials

We show that the sum of the four parameter weights over all ordinary or strict partitions with parts each less than or equal to a given integer $N$ is expressed by the Al-Salam Chihara polynomials. This weight is a generalization of the Andrews-Stanley partition function. As a corollary we prove C. Boulet's results when the sum runs over all ordinary or strict partitions. In the last section we study the weighted sum of Schur's $P$-functions and $Q$-functions, where the sum runs over all strict partitions.

$(q,y)$-Laguerre 多项式及其矩的组合学

Combinatorics of $(q,y)$-Laguerre polynomials and their moments

本文研究 $(q,y)$-Laguerre 多项式，证明其为 Al-Salam–Chihara 多项式的缩放形式，并通过彩色 Laguerre 构型及排列统计量给出组合解释与矩描述，同时证明线性化系数具有非负整系数。

The combinatorics of q-Charlier polynomials

We describe various aspects of the Al-Salam-Carlitz $q$-Charlier polynomials. These include combinatorial descriptions of the moments, the orthogonality relation, and the linearization coefficients.

Around Poisson--Mehler summation formula

We study polynomials in $x$ and $y$ of degree $n+m:\allowbreak \{Q_{m,n}(x,y|t,q)\}_{n,m\geq 0}$ that appeared recently in the following identity: $γ_{m,n}(x,y|t,q) \allowbreak =\allowbreak γ_{0,0}(x,y|t,q) \allowbreak Q_{m,n}(x,y|t,q) $ where $γ_{m,n}(x,y|t,q) \allowbreak =\allowbreak \sum_{i\geq 0}\frac{t^{i}}{[i]_{q}}H_{i+n}(x|q) H_{m+i}(y|q)$, $\allowbreak $ $\{H_{n}(x|q)}_{n\geq -1}$ are the so-called $q-$% Hermite polynomials (qH). In particular we show that the spaces $span\{Q_{i,n-i}(x,y|t,q) :i=0,...,n\}_{n\geq 0}$ are orthogonal with respect to a certain measure (two-dimensional $(t,…