三类WZ-方程的一些探讨
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摘要
本文综合考虑以下三类相互区别但又具备相似性的函数方程的求解与一些基本应用.此处△x,(?),Dq,x依次表示关于x的差分、偏导与q差分,故相应地被称为离散型、连续型与q型WZ函数方程.所用方法就是生成函数法(或称为形式幂级数法).主要内容是：
第一章对上述三种类型的函数方程做了一个简短介绍.
第二章主要推导出生成函数与离散型WZ方程的关系,并从生成函数角度给出超几何恒等式的新的证明方法,同时也从生成函数角度推导出WZ对偶之间(新的)关系等式
本文第三章主要讨论连续型WZ函数方程及广义的连续型WZ函数方程及其解在化简含参变量积分,含参变量积分的渐近估计,由含参变量积分所定义的函数的定积分计算,和含参变量积分的D'Alembert函数表示等问题.其中有些问题由华南师大的陈奕俊率先提出和研究[7,8,9,10].我们主要是在求解连续型WZ函数方程,特殊的连续型WZ函数方程,及一般情形下连续型WZ函数方程的解.然后结合陈的主要结果,推导一般情形的连续型WZ函数方程下的含参变量积分化简定理,并求出两个无穷限反常积分.
本文最后部分基于华东师大的刘治国q算子等式的最新工作[17]而展开讨论.刘的q算子方法可以给出基本超几何级数理论许多经典结果的优美证明.其中关键性结论可归纳为上述的q型WZ函数方程.利用生成函数方法,可以给出该种类型方程的一般解(它包含刘治国的结果).同时从中可以给出关于Rogers-Szego多项式的q-Mehler公式的初等证明.
This present thesis is concerned with the following three kindsof diferent but similar WZ-type functional equations in a unified view-point
together 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–Zeilberge
Chapter 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.
第一章对上述三种类型的函数方程做了一个简短介绍.
第二章主要推导出生成函数与离散型WZ方程的关系,并从生成函数角度给出超几何恒等式的新的证明方法,同时也从生成函数角度推导出WZ对偶之间(新的)关系等式
本文第三章主要讨论连续型WZ函数方程及广义的连续型WZ函数方程及其解在化简含参变量积分,含参变量积分的渐近估计,由含参变量积分所定义的函数的定积分计算,和含参变量积分的D'Alembert函数表示等问题.其中有些问题由华南师大的陈奕俊率先提出和研究[7,8,9,10].我们主要是在求解连续型WZ函数方程,特殊的连续型WZ函数方程,及一般情形下连续型WZ函数方程的解.然后结合陈的主要结果,推导一般情形的连续型WZ函数方程下的含参变量积分化简定理,并求出两个无穷限反常积分.
本文最后部分基于华东师大的刘治国q算子等式的最新工作[17]而展开讨论.刘的q算子方法可以给出基本超几何级数理论许多经典结果的优美证明.其中关键性结论可归纳为上述的q型WZ函数方程.利用生成函数方法,可以给出该种类型方程的一般解(它包含刘治国的结果).同时从中可以给出关于Rogers-Szego多项式的q-Mehler公式的初等证明.
This present thesis is concerned with the following three kindsof diferent but similar WZ-type functional equations in a unified view-point
together 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–Zeilberge
Chapter 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.
引文
[1]S. A. Abramov, M. Petkovsek. D'Alembertian Solutions of Linear Differential and Difference Equations. Proc. ISSAC,94:169-174.380:185-198.
[2]G. Almkvist, D. Zeilberger. The method of differentiating under the integral sign. J of Symbolic Computation,10(1990)571-591.
[3]T. Amdeberhan. Faster and faster convergent series for ζ(3). Elect of Combin,1996,3:R13.
[4]D. Bleecker, G. Csordas,李俊杰译.基础偏微分方程.高等教育出版社,2006.
[5]J. Borwen, D. Baley. Mathematics by experiment:Plausible reasoning in the21st century. Massachusetts:A K Peters, Ltd,2004.
[6]陈奕俊.WZ方法与一类组合和的渐近估计问题.华南师范大学学报：自然科学版,2009(1)1-5.
[7]陈奕俊.WZ方法与一类含参变量积分的渐近估计问题.华南师范大学学报：自然科学版,5463(2009)1-6.
[8]陈奕俊.WZ方法与一类定积分的计算及推广.华南师范大学学报：自然科学版,5463(2011)29-35.
[9]陈奕俊.推广的Leibniz法则,Abramov-Petkovsk算法与一类含参变量积分的D'Alembert函数表示.待发表.
[10]陈奕俊.WZ方法与一类含参变量积分所定义的函数的定积分问题.待发表.
[11]初文昌,王晓霞Basic Hypergeometric Series-Quick Access to Identities. Journal of Mathematical Research and Exposition,(2008)223-250.
[12]费定晖.吉米多维奇数学分析习题集(五).山东科学技术出版社,2007.
[13]G. Gasper, M.Rahman. Basic Hypergeometric Series. Cambridge University Press,2004.
[14]I. S. Gradshteyn, I. M. Ryzhik. Table of Intergrals Series and Products.6th ed New York:Academic Press,2000.
[15]J. Gullera. Some binomial series obtained by the WZ method. Advances in Applied Mathematics,29(2002)599-603.
[16]F. H. Jackson. q-Difference equations. America J. Math.,32(1910)305-314.
[17]Zhi-Guo Liu. Two q-difference equations and q-operator identities. Journal of Difference Equations and Applications,(2010)1293-1307.
[18]M. Petkovsek, H. S. Wilf, and D. Zeilberger. A=B. A. K. Peters Wellesly,1996.
[19]H. S. Wilf. Accelerated series for universal constants by WZ method. J of Discrete Mathematics and Theoretical Computer Science,3(1999)189-192.
[20]H. S. Wilf, D. Zeilberger. An algorithmic proof theory for hypergeomet-ric (ordinary and "q") multisum/integral identities. Inventiones Mathematica,108(1992)575-633.
[21]D. Zelberger. Closed form (pun intended!). Contemporary Mathematic,70(1993)537-566.
[22]D. Zelberger, T. Amdeberhan. Hypergeometric series accelration via the WZ method. Elect J of Combin,1997,4:R3.
[23]钟玉泉.复变函数论(第三版).高等教育出版社,2004.
[2]G. Almkvist, D. Zeilberger. The method of differentiating under the integral sign. J of Symbolic Computation,10(1990)571-591.
[3]T. Amdeberhan. Faster and faster convergent series for ζ(3). Elect of Combin,1996,3:R13.
[4]D. Bleecker, G. Csordas,李俊杰译.基础偏微分方程.高等教育出版社,2006.
[5]J. Borwen, D. Baley. Mathematics by experiment:Plausible reasoning in the21st century. Massachusetts:A K Peters, Ltd,2004.
[6]陈奕俊.WZ方法与一类组合和的渐近估计问题.华南师范大学学报：自然科学版,2009(1)1-5.
[7]陈奕俊.WZ方法与一类含参变量积分的渐近估计问题.华南师范大学学报：自然科学版,5463(2009)1-6.
[8]陈奕俊.WZ方法与一类定积分的计算及推广.华南师范大学学报：自然科学版,5463(2011)29-35.
[9]陈奕俊.推广的Leibniz法则,Abramov-Petkovsk算法与一类含参变量积分的D'Alembert函数表示.待发表.
[10]陈奕俊.WZ方法与一类含参变量积分所定义的函数的定积分问题.待发表.
[11]初文昌,王晓霞Basic Hypergeometric Series-Quick Access to Identities. Journal of Mathematical Research and Exposition,(2008)223-250.
[12]费定晖.吉米多维奇数学分析习题集(五).山东科学技术出版社,2007.
[13]G. Gasper, M.Rahman. Basic Hypergeometric Series. Cambridge University Press,2004.
[14]I. S. Gradshteyn, I. M. Ryzhik. Table of Intergrals Series and Products.6th ed New York:Academic Press,2000.
[15]J. Gullera. Some binomial series obtained by the WZ method. Advances in Applied Mathematics,29(2002)599-603.
[16]F. H. Jackson. q-Difference equations. America J. Math.,32(1910)305-314.
[17]Zhi-Guo Liu. Two q-difference equations and q-operator identities. Journal of Difference Equations and Applications,(2010)1293-1307.
[18]M. Petkovsek, H. S. Wilf, and D. Zeilberger. A=B. A. K. Peters Wellesly,1996.
[19]H. S. Wilf. Accelerated series for universal constants by WZ method. J of Discrete Mathematics and Theoretical Computer Science,3(1999)189-192.
[20]H. S. Wilf, D. Zeilberger. An algorithmic proof theory for hypergeomet-ric (ordinary and "q") multisum/integral identities. Inventiones Mathematica,108(1992)575-633.
[21]D. Zelberger. Closed form (pun intended!). Contemporary Mathematic,70(1993)537-566.
[22]D. Zelberger, T. Amdeberhan. Hypergeometric series accelration via the WZ method. Elect J of Combin,1997,4:R3.
[23]钟玉泉.复变函数论(第三版).高等教育出版社,2004.