Some Discussions On Three Kinds Of WZ–equations

This present thesis is concerned with the following three kindsof diferent but similar WZ-type functional equations in a unified view-pointtogether with some basic applications. Here, x,, Dq,xdenote respec-tively diferent, derivative, and q–diference with respective to a variablex. They are called correspondingly discrete, continuous, and q-WZ equa-tions. The main tool is the generating function (or formal power series).More precisely, the main content includes the following:Chapter one is devoted to a brief introduction to three kinds ofWZ-type equations.In Chapter two, we reformulate the discrete WZ-equation in termsof generating function. Consequently, some known results such as com-panion identities and dual identities can be expressed in a short and newmanner. One of main results is the following companion identity due toWilf–ZeilbergeChapter three is devoted to the problems of approximating esti-mate, commutating of definite integral, as well as the representation of D’Alembert function defined by certain parametric integrals. All aresubject to the continuous and generalized and continuous WZ-equationIt should be mentioned that some of these problems are first posed andconsidered in [7,8,9,10]. Our main purpose is to solve continuous andgeneralized and continuous WZ–equations under various conditions andto specialize the main theorem of Chen. Some new results are obtained.The last part of the paper is based on the recent work of Liu [17]on q–diference formula. Certainly, the method of q–diference leads usto some surprising proofs of many summation and transformation for-mulas in q–series theory. One of the most crucial fact is q–WZ–equationabove. By means of generating functions, we find the general solutionof such sort containing Liu’s result as a specific case. Also, an new butelementary proof of the q–Mehler formula for Rogers-Szeg¨o polynomialscan also derived from our argument.