Husam L. Saad

The bivariate Rogers-Szegö polynomials

We present an operator approach to deriving Mehlers formula and the Rogers formula for the bivariate Rogers–Szego polynomials hn(x, y|q). The proof of Mehlers formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big q-Hermite polynomials Hn(x; a|q) due to Askey, Rahman and Suslov. Mehlers formula for hn(x, y|q) involves a 32 sum and the Rogers formula involves a 21 sum. The proofs of these results are based on parameter augmentation with respect to the q-exponential operator and the homogeneous q-shift operator in two variables. By extending recent results on the Rogers–Szego polynomials hn(x|q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for hn(x, y|q). Finally, we give a change of base formula for Hn(x; a|q) which can be used to evaluate some integrals by using the Askey–Wilson integral.

An operator approach to the Al-Salam–Carlitz polynomials

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–Szego polynomials.

On the Gosper-Petkovšek representation of rational functions

We show that the uniqueness of the Gosper-Petkovsek representation of rational functions can be utilized to give a simpler version of Gospers algorithm. This approach also applies to Petkovseks generalization of Gospers algorithm, and its q-analogues by Abramov-Paule-Petkovsek and Boing-Koepf.