The Applications And Extensions Of Q-difference Equations And High Dimensional Q-integrals

In recent years,basic hypergeometric series has been developing rapidly,but it is limited in studying the form of q-orthogonal polynomials,finite summation formulas and integral operations.But it can solve this problem effectively from the perspective of q-difference equation.This paper is divided into three chapters to introduce how to use the q-differential equation,q-integral,q-orthogonal poly-nomials and some basic computational methods to study basic hypergeometric field of some commonly used formula and the promotion and expansion of it.Firstly,by using some properties of q-homogeneous operator,we construc-t the bivariate Cigler polynomials,and get the q-difference equation related to them.Then we get the generating function of Cigler polynomials,then we use the homogeneous q-difference equation.The Andrew-Askey integral,the Askey-Roy integral,the moment integral and the U(n+1)formula are generalized,and four identities are obtained.Secondly,using the definition of q-fractional order integral obtained double fractional order q-integral formula,and by using the definition of polynomial is obtained for a class of double fractional order q-integral,the first application of fractional q-more points,and then through the research and application of bilinear,three linear polynomial of the generating function of fractional q-double integral.Finally,based on the definition of q-integral formula defined by Thomae-Jackson and the research of Liu's double q-integral formula,we constructed a class of three fold q-integral formula,and got some equations.Four equations are obtained by the study of the q-Chu-Vandermonde formula and the U(n+1)formula.