Abstract
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-Szegő polynomials and enables us to develop a systematic method to prove the identities involving the Rogers-Szegő 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-Szegő 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.
| Original language | English |
|---|---|
| Pages (from-to) | 1199-1216 |
| Number of pages | 18 |
| Journal | Science China Mathematics |
| Volume | 66 |
| Issue number | 6 |
| DOIs | |
| State | Published - Jun 2023 |
Keywords
- 05A30
- 05A40
- 33D15
- 33D99
- Rogers-Szegő polynomial
- q-derivative
- q-exponential operator
- q-operational equation
- q-series