Q-Orthogonal Polynomials And Related Problems

(1) New proofs and generalizations of the family of generating functions for Rogers-Szego polynomials are given. First, the relations between exponential op-erators and Rogers-Szego polynomials are built, then new proofs of generating functions and their dual forms are obtained from the perspective of operators. In addition, generating functions for bivariate Rogers-Szego polynomials are stud-ied by constructing the generalized exponential operators, then an open problem is proposed. At last, the U(n+1)-type generating functions for Rogers-Szego polynomials and U(n+1)-type transformations of Kalnins-Miller are further con-sidered.(2) News proofs of two-term expansions of the multilinear generating func-tions for Rogers-Szego and Hahn polynomials are given. First, Al-Salam-Carlitz's orthogonal polynomials and their related integrals are derived by relations be-tween moments and orthogonal polynomials. In addition, two-term expansions of the bilinear generating functions for Hahn polynomials are deduced. Further-more, two-term expansions of trilinear and multilinear generating functions for Hahn polynomials are researched. At last, the related results among moments, Euler's finite differences and Carlitz's inversions are achieved.(3) The results of multivariable q-Laguerre polynomials are gained. First, the characters of univariate q-Laguerre polynomials are discussed by utilizing the orthogonality. In addition, the relations between generalized q-Hermite polynomi-als and q-Laguerre polynomials are presented. Furthermore, the mixed integrals of q-Hermite and q-Laguerre polynomials are investigated. At last, the orthog-onality of discrete integrals involving multivariable q-Hermite polynomials are extended.(4) The multiple generalizations of generating functions for Carlitz-type poly-nomials and Christoffel-Darboux formulas are shown. First, Carlitz-type polyno-mials are studied by using the technique of decomposed exponential operators. In addition, the Christoffel-Darboux formulas of Rogers-Szego polynomials are re-searched. Furthermore, some results of Carlitz are modified. At last, a q-analogue of binomial theorem is given by using Carlitz's q-operators.(5) The asymptotic property of ratio type gamma function and the convexity of psi function are studied. First, an inequality of ratio type gamma function is improved by using logarithmically completely monotonicity. In addition, the property of psi function are researched by convolution for Laplace transform, some corresponding byproducts are derived.