A note on generalized $q$-difference equations for general Al-Salam--Carlitz polynomials

An Operator Approach to the Al-Salam-Carlitz Polynomials

We present an operator approach to Rogers-type formulas and Mehler's formulas for the Al-Salam-Carlitz polynomials $U_n(x,y,a;q)$. By using the q-exponential operator, we obtain a Rogers-type formula which leads to a linearization formula. With the aid of a bivariate augmentation operator, we get a simple derivation of Mehler's formula due to by Al-Salam and Carlitz, which requires a terminating condition on a ${}_3ϕ_2$ series. By means of the Cauchy companion augmentation operator, we obtain Mehler's formula in a similar form, but it does not need the terminating condition. We also give sever…

多变量阿尔 - 萨拉姆 - 卡尔itz 多项式的双线性生成函数及其应用

Bilinear generating functions of the multivariable Al-Salam-Carlitz polynomials and applications

研究多变量阿尔 - 萨拉姆 - 卡尔itz 多项式，通过比较系数法建立新生成函数，其含特殊公式，还给出相关新\(q -\)级数恒等式。

具有 Al-Salam–Carlitz 权重的 $q$-形变酉系综的谱分析

Spectral analysis of $q$-deformed unitary ensembles with the Al-Salam--Carlitz weight

本文研究带 Al-Salam–Carlitz 权重的 $q$-形变随机酉系综，借助 Flajolet–Viennot 理论与匹配组合学得到谱矩正和式及大 $N$ 展开，并给出含两次软边缘相变的极限谱密度。

关于Rogers-Szegö多项式$q$-微分算子的一点注记

A Note on the Rogers-Szegö Polynomial $q$-Differential Operators

本文引入Rogers-Szegö变形$q$-微分算子，其极限为Chen的$q$-指数算子，并利用该算子表示变形齐次Al-Salam-Carlitz多项式，同时给出新一般Al-Salam-Carlitz多项式与基本超几何级数的关系恒等式。

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.

A note on log-convexity of q-Catalan numbers

The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefficients. They evaluate, at q=1, to the Catalan numbers: 1, 1, 2, 5, 14,..., a log-convex sequence. We use a combinatorial interpretation of these polynomials to prove a q-log-convexity result. The sequence of q-Catalan numbers is not q-log-convex in the narrow sense used by other authors, so our work suggests a more flexible definition of q-log convex be adopted.