The Application Research Of Q-difference Equations In Moment Integral And Q-polynomials

This article's main research is building the relationships between q-difference equations and q-orthogonal polynomials,than by the method of reconstructing and expanding,we obtains new q-difference equations,which are combined with a series of well-known hypergeometric series identities to derive new identities.The main content of this article are:1.According to the definition of moments integral,we use the definition to express the q-orthogonal polynomial in the form of integral of moments integral,and than combine the q-differential equation to derive two new identities of moments integrals,which are connected in parallel with Rogers-Szego polynomials,Hahn polynomials,and Al-Salam-Carlitz polynomials,getting more new generating functions of q-orthogonal polynomials.2.We have studied the recent results of Arjika,and defined two new q-exponential operators based on the results of Arjika's research,and then calculated the new generalized Al-Salam-Carlitz polynomials from these two newly defined operators and gave the corresponding q-difference equation.We connect the new q-difference equation with the transformation identity,and give the corresponding transformation identities after a series of operations.We also apply the q-difference equation and the q-exponent operator to the q-polynomial In the U(n+1)type generating function of and the Ramanujan's integral,more complex identities are expanded.3.We studied the generalized Al-Salam-Carlitz polynomial and the(q,c)-exponent operator,and introduced two new(q,c)-exponent operators,which can form new polynomials,namely generalized(q,c)-Al-Salam-Carlitz polynomials,and expand new ones based on the connection between q-polynomials and q-exponential operators(q,c)difference equation.Using the new(q,c)-difference equation and(q,c)-exponent operator,give the generating function of the(q,c)-Al-Salam-Carlitz polynomial.Then,applying the main(q,c)-difference equation theorem,generalizing the Ramanujan's integral,q-Chu-Vandermonde formula and Watson contour integral,and obtaining many q-series identities.