The Applications Of Q-difference Equations And Fractional Q-calculus

For the last few years,q-series has become the research object of more and more scholars,basic hypergeometric series also got rapid developmenty.However,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 the following three parts to introduce how to use the q-differential equation and some basic calculation methods to study the Askey-Wilson integral,Fractional q-polynomial,q-Hermite polynomial,and get some operator identities and generating functions,at the same time,the corresponding generating functions are promoted and extended Firstly,In this paper,q-difference equations and related problems in special functions,whose formal solutions are q-polynomials,are discussed.As well as Askey-Wilson integral and its inverse integral are extended by the method of the q-difference equation.In addition,Bailey 6ψ6 summation is generalized by q-difference equationSecondly,In this paper,we establish the relation between fractional q-integral and generating functions for q-polynomial,we also give the definition generalized Predrag-Sladjana-Miomir polynomials.In addition,we discuss generating func-tions for solutions of q-difference equation.Moreover,we deduce some generating functions for generalized Predrag-Sladj ana-Miomir polynomialsFinally,Based on the binary q-Hermite polynomials studied by Ismail and Zhang,We constructed ternary q-Hermite polynomials and their dual forms,we constructed the relation between ternary q-Hermite polynomials and q-differential operators.We find a suitable partial q-differential equation,and by using the prop-erties of the equation,we get the generating function of the ternary q-Hermite polynomials,Srivastava-Agarwal generating functions,Nielsen generating func-tions and mixed generating functions.