The bivariate Rogers–Szegö polynomials

William Y C Chen, Husam L Saad, Lisa H Sun
2007 Journal of Physics A: Mathematical and Theoretical  
We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szegö polynomials h_n(x,y|q). The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big q-Hermite polynomials H_n(x;a|q) due to Askey, Rahman and Suslov. Mehler's formula for h_n(x,y|q) involves a _3ϕ_2 sum and the Rogers formula involves a _2ϕ_1 sum. The proofs of these results are based on parameter augmentation
more » ... ith respect to the q-exponential operator and the homogeneous q-shift operator in two variables. By extending recent results on the Rogers-Szegö polynomials h_n(x|q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for h_n(x,y|q). Finally, we give a change of base formula for H_n(x;a|q) which can be used to evaluate some integrals by using the Askey-Wilson integral.
doi:10.1088/1751-8113/40/23/005 fatcat:yvbe6bye6bgyrnseyjxrej5j7a