Abstract:  We obtain Mehler's formula and the Rogers formula for the continuous big q-Hermite polynomials H_n(x; a|q). Instead of working with the polynomials H_n(x; a|q) directly, we consider the equivalent forms in terms of the bivariate Rogers-Szegö polynomials h_n(x; y|q) recently introduced by Chen, Fu and Zhang. It turns out that Mehler's formula for H_n(x; a|q) involves a ₃φ₂ sum, and the Rogers formula involves a ₂φ₁ 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.

  AMS Classification:  05A30, 33D45.

  Keywords:  The bivariate Rogers-Szegö polynomials, the continuous big q-Hermite polynomials, the Cauchy polynomials, the q-exponential operator, the homogeneous q-shift operator

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