Relative Research On Q-difference Equations, Q-operator Identities And Q-polynomials

In this paper, we study some relative problems in q-operator methods, Bailey transformation, inversion relation and q-difference equations on special function. We will point out that q-operator identities and q-difference equations are tools to generalize some known results. Moreover, Bailey transformation and inversion methods are very important tools in combinatorics and special function theo-ry, respectively. However, we get some new results by using them. Specially. Nassrahall-Rahman integral has been proved to be very useful to q-integrals by taking different methods. We will give a general q-beta integral to investigate Nassrahall-Rahman integral. In accordance with section order, the main results are as follows:Using the q-Chu-Vandermonde formula, we prove two new q-exponential operator identities, which are the extensions of Chen-Liu’s results. These two q-exponential operator identities are used to derive some new q-summation for-mulas and q-integrals.We use the q-difference equations to investigate some properties of the q-polynomials generating functions. Firstly, we introduce two new q-polynomials ψn(a,c) (x, y|q) and sn(a,c)(x, y, z|q). Al-Salam-Carlitz polynomials ψn(a) (x, y|q) and Saad-Sukhi polynomials sn(x,y,b;q) are the special cases of new q-polynomials, respectively. Also we derive several types of generating functions for new q-polynomials by solving two simple q-difference equations. In the scope of q-difference equations, we get some generalizations of Andrews-Askey integral which includes some known-results. In particular, we give a new proof for the gener-ating function, the Mehlers formula for Un(x,y,a;q). Moreover, we give a con-jecture of the multivariable function expansion, which is derived from the two-variable function expansion about -operator identities, q-difference equations and q-polynomials.Combining Bailey transform and Bressoud inversion, we prove a general ex-pansion formula in Askey-Wilson polynomials. As applications, we give new proofs and generalize some recent results by Ismail-Stanton and Liu. More-over, we prove a new q-beta integral formula involving Askey-Wilson polynomials, which also includes the Nassrallah-Rahman integral as a special case. In addition, we give a bootstrapping proof of Ismail-Stanton’s recent generating function of Askey-Wilson polynomials.Motivated by results of Warnaar, we formulate two new Bailey lattices. We describe a number of new q-multisums with multiple variables for basic hyper-geometric series which are derived from consequences of these two new Bailey lattices. As applications, we derive two new transformations for basic hyperge-ometric by using the unit Bailey pair. Moreover, these Bailey lattices could be used to get some mock theta functions.