  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 ${}_3\phi_2$ 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 several identities
on the generating functions for products of the Al-Salam-Carlitz polynomials
which are extensions of formulas for Rogers-Szeg\"o polynomials.

Classical Analysis and ODEs;Combinatorics