| With the rapid development of nonlinear mathematics and quantum math-ematics,in the combinatorial mathematics,complexity of integral operation and finiteness of summation formula are important factors which restrict the progress of research.In this paper,we structure q-difference equation model with the formal solution of q-exponential operator,use the way of getting the difference equation form solution to generalize Sears formula,Al-Salam-Carlitz polynomial generating functions,Andrews-Askey integral,Heine formula and so on.This paper is divided into three chapters:One,we construct a new q-operator identity,and link it with a q-difference equation.Frist,combing with definitions of Dq,θq andtheir function characteris-tics,we construct.a new q-operator identity.Secondly,finding the correspending q-difference equation,we connect the new q-operator identity with the formal so-lution of the q-difference equation.Finally,we expand the relation of q-operator and its difference equation to another q-operator with two variables,and then we get a set of q-difference equations.Two,we give the expansions of q-Chu-Vandermonder,s formula,Sears’s for-mula,Anderws-Asley’s integral and the generating function of Al-SalamCarlitz’s polynomials.Frist,we use the q-dislocation that restructed to promote q-Chu-Vandermonde,s formula,Sears’s formula,Anderws-Asley’s integral.Secondly,we give new proofs these promotions with the help of difference equation’s form.Three,in the context.of homogeneous operator of Dxy and θxy,we give some applications of homogeneous q-difference equation’s form solution,and then we prove Euler’ formula,Rogers-Szeg? polynomial with three variables and Sn(x,y,z|q)polynomial. |
