广义Al-Salam-Carlitz $q$ -多项式族的Generating Functions
Generating Functions for Some Families of the Generalized Al-Salam-Carlitz $q$-Polynomials
摘要：
本文利用人们熟悉的$q$ -差分算子$D_q$和$D_{q^{-1}}$，首先引入了两个齐次$q$ -差分算子$\mathbb{T}({\bf a},{\bf b},cD_q)$和$\mathbb{E}({\bf a},{\bf b}, cD_{q^{-1}})$，这两个算子适合于处理广义Al-Salam-Carlitz $q$ -多项式族$ϕ_n^{({\bf a},{\bf b})}(x,y|q)$和$ψ_n^{({\bf a},{\bf b})}(x,y|q)$ .然后，我们应用这两个齐次$q$ -差分算子，以获得生成函数，罗杰斯型公式，扩展的罗杰斯型公式和Srivastava-Agarwal型线性以及双线性生成函数涉及这些家庭的广义Al-Salam-Carlitz $q$ -多项式。我们还展示了如何在这里提出的各种结果与那些在许多早期的作品，我们在本文中研究的主题。
英文摘要：
In this paper, by making use of the familiar $q$-difference operators $D_q$ and $D_{q^{-1}}$, we first introduce two homogeneous $q$-difference operators $\mathbb{T}({\bf a},{\bf b},cD_q)$ and $\mathbb{E}({\bf a},{\bf b}, cD_{q^{-1}})$, which turn out to be suitable for dealing with the families of the generalized Al-Salam-Carlitz $q$-polynomials $ϕ_n^{({\bf a},{\bf b})}(x,y|q)$ and $ψ_n^{({\bf a},{\bf b})}(x,y|q)$. We then apply each of these two homogeneous $q$-difference operators in order to derive generating functions, Rogers type formulas, the extended Rogers type formulas and the Srivastava-Agarwal type linear as well as bilinear generating functions involving each of these families of the generalized Al-Salam-Carlitz $q$-polynomials. We also show how the various results presented here are related to those in many earlier works on the topics which we study in this paper.
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