Research On Some Q-expansion Formulas And Their Applications

In this paper,we mainly start from the three methods of q-Taylor expansion,Liu's Calculus and q-Exponential operator expansion,and study the q-expansion formulas based on(q,?)-differential operator ?Dq.Firstly,we give the Lagrangian expansion formula about ?Dq.Secondly,we obtain the expansion theorem about the Generalized Rogers-Szeg? polynomials ??Hn(x,y|q)and Generalized Stieltjes-Wigert polynomials??Gn(x,y|q)by using the method of Liu's Calculus.Thirdly,we obtain the expansion formulas about(q,?)-Exponential operator and Generalize some of Zhang Wenjing's conclusions by using the method of q-Exponential operator expansion.Finally,we discuss some exceptional cases and combine the generating functions of the classical Rogers-Szego polynomials and Al-Salam-Carlitz polynomial and its q-Mehler formula to get some new identities.