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

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