An Operator Approach to the Al-Salam-Carlitz Polynomials
William Y. C. Chen, Husam L. Saad, and Lisa H. Sun
Abstract: We present an operator approach to Rogers-type formulas and Mehler's formula for the Al-Salam–Carlitz polynomials Un(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 Al-Salam and Carlitz ["Some orthogonal q-polynomials," Math. Nachr. 30, 47 (1965)]. By means of the Cauchy companion augmentation operator, we obtain an equivalent form of Mehler's formula. We also give several identities on the generating functions for products of the Al-Salam-Carlitz polynomials, which are extensions of the formulas for the Rogers–Szegö polynomials.
AMS Classification: 33D45, 05A30