A q-operational equation and the Rogers-Szego polynomials

作者：Liu, Zhiguo^*

来源：Science China Mathematics, 2023, 66(6): 1199-1216.

DOI：10.1007/s11425-021-1999-2

• q-series
• q-derivative
• q-operational equation
• q-exponential operator
• Rogers-Szego polynomial

By solving a q-operational equation with formal power series, we prove a new q-exponential operational identity. This operational identity reveals an essential feature of the Rogers-Szego polynomials and enables us to develop a systematic method to prove the identities involving the Rogers-Szego polynomials. With this operational identity, we can easily derive, among others, the q-Mehler formula, the q-Burchnall formula, the q-Nielsen formula, the q-Doetsch formula, the q-Weisner formula, and the Carlitz formula for the Rogers-Szego polynomials. This operational identity also provides a new viewpoint on some other basic q-formulas. It allows us to give new proofs of the q-Gauss summation and the second and third transformation formulas of Heine and give an extension of the q-Gauss summation.