  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 $\phi_n^{({\bf a},{\bf
b})}(x,y|q)$ and $\psi_n^{({\bf a},{\bf b})}(x,y|q)$. 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.
