The Applications Of Q-difference Equation Formula On Q-integral

In recent years,the rapid development of basic hypergeometric series,but the form of q-orthogonal polynomials and the sum formula and integral operation are limited.From the point of view of q-difference equation,the problem can be solved effectively.This paper is divided into three chapters to introduce some identities which in basic hypergeometric fields by q-operator,q-difference equation and the recursive formula of Al-Salam-Verma integral.First,The construction of q-exponent operator and it's dual operator are respectively used for fractional,we obtained q-polynomials,And the q-difference equation is obtained,we extended q-Cauchy binomial theorem and Andrews q-beta integrals,then we obtained two new basic identities.Second,we give two q-difference equations which has five paramet.er by the new q-exponential operator identities,then extended many important formula of basic hypergeometric series.As applications,we redefined Ramanujan's formulas for q-beta integrals,the generalization of Andrews-Askey integrals are obtained.We also obtained the generating function for generalized Al-Salam-Carlitz poly-nomials.Third,we give Al-Salam-Verma integral formula by the deformation of q-Chu-Vandermonde formula and the method of q-difference equations,In addition,a recurring formula of Al-Salam-Verma integral are given by the method of cyclic iterative,besides,a new transformation identity is gained by the generalization of Al-Salam-Verma integral.At last,we give a new proof of Gesper's formula which is a generalization of Askey-Roy integral.