Some Q-orthogonal Polynomials And Q-partial Differential Equations

We study the problem of expanding an analytic function in terms of several q-orthogonal polynomials.We define a new q-differential(derivative)operator.1.We establish the relation between Al-Salam-Carlitz polynomials and q-partial differential equations.we prove that an analytic function can be expanded in terms of homogeneous generalized Al-Salam-Carlitz polynomials if and only if the analytic function satisfies certain q-partial differential equations.As appli-cations,we obtain the bilinear generating functions and multilinear generating functions for the homogeneous generalized Al-Salam-Carlitz polynomials,further-more we generalize Andrews-Askey integral and Ramanujan integral.At last,we derive U(n + 1)type generating functions for the homogeneous generalized Al-Salam-Carlitz polynomials.2.We introduce a new generalized q-differential(derivative)operator,which include the two classical q-differential operators as special cases,then we give the Leibniz formula for the new q-differential operator.Using this operator,we study q-Laguerre polynomials which with a more complicate form.First,we define a class of q-Laguerre type polynomials,which include q-Laguerre polynomials,little q-Laguerre polynomials and q-Hahn polynomials as special cases.With the new established q-differential operator,we investigate the relation between the expanding of an analytic function in terms of q-Laguerre polynomials and q-partial differential equations.We derive a q-partial differential equation with new q-differential operator.The generality of this operator make it can be used freely to the q-Laguerre polynomials.This could not work when using with the classical q-differential operator.Using these results,we obtained the generating functions,bilinear generating functions and mixed generating functions for q-Laguerre polynomials.Some q-integrals with q-Laguerre polynomials are also obtained.At last,we introduce a class of ternary q-Laguerre polynomials,which include 2D q-Laguerre polynomials and 2D little q-Laguerre polynomials given by Ismail et al,also using the new established q-differential operator,we obtain the sufficient and necessary condition of an analytic function expanding in terms of ternary q-Laguerre polynomials.3.We introduce a class of ternary q-Hermite polynomials,which include many specific polynomials,such as the two q-Hermite polynomials given by Ismail and Zhang[49]and other polynomials.An inverse expressions of Hermite polyno-mials are obtained.We study the necessity and sufficiency of an analytic function expanded in terms of ternary q-Hermite polynomials,the new q-differential oper-ator is also involved.At last,we obtain several generating functions for ternary q-Hermite polynomials,some of which generalize the corresponding results in [49].