An Operator Approach to the Al-Salam-Carlitz Polynomials

Abstract

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 ${}_3{\varphi }_2$ 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-Szegö polynomials.

15 pages

References in corpus (3)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• The q-harmonic oscillator and the Al-Salam and Carlitz polynomials
• The Bivariate Rogers-Szegö Polynomials