A q-operational equation and the Rogers-Szegő polynomials  

作　　者：Zhiguo Liu 

机构地区：[1]School of Mathematical Sciences,East China Normal University,Shanghai 200241,China

出　　处：《Science China Mathematics》2023年第6期1199-1216,共18页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No.11971173);Science and Technology Commission of Shanghai Municipality (Grant No.13dz2260400)。

摘　　要：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.

关 键 词：Q-SERIES Q-DERIVATIVE q-operational equation q-exponential operator Rogers-Szegőpolynomial 

分 类 号：O175[理学—数学]