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 \(\mathbb{T}(\mathbf{a},\mathbf{b},cD_{q})\) and \(\mathbb{E}(\mathbf{a},\mathbf{b}, cD_{q^{-1}})\), which turn out to be suitable for dealing with the families of the generalized Al-Salam–Carlitz q-polynomials \(\phi_{n}^{(\mathbf{a},\mathbf{b})}(x,y|q)\) and \(\psi_{n}^{(\mathbf{a},\mathbf{b})}(x,y|q)\). 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.

在本文中，通过使用熟悉的q差分算子\（D_ {q} \）和\（D_ {q ^ {-1}} \），我们首先介绍两个齐次的q差分算子\（\ mathbb {T}（\ mathbf {a}，\ mathbf {b}，cD_ {q}）\）和\（\ mathbb {E}（\ mathbf {a}，\ mathbf {b}，cD_ {q ^ {- 1}}）\），它适合处理广义的Al-Salam–Carlitz q-多项式\（\ phi_ {n} ^ {（\ mathbf {a}，\ mathbf {b} ）}（x，y | q）\）和\（\ psi_ {n} ^ {（\ mathbf {a}，\ mathbf {b}）}（x，y | q）\）。然后，我们应用这两个齐次q的每一个-差分算子，以生成生成函数，Rogers型公式，扩展的Rogers型公式以及Srivastava-Agarwal型线性和双线性生成函数，这些函数涉及广义Al-Salam-Carlitz q多项式的每个族。我们还展示了此处提出的各种结果与我们在本文中研究的主题的许多较早著作中的结果如何相关。