超几何函数与分数阶积分算子

发布时间：2018-05-30 01:10

本文选题：分数阶积分算子 + 分数阶分部积分　；参考：《华东师范大学》2017年博士论文

【摘要】：本文的主要目的是为广义分数阶积分算子I和J建立一套新的理论.这两个算子的特征是其积分核中均包含一类十分特殊的广义超几何函数r+2Fr+1.同时,算子I和J的重要性也正是源于它们所包含的这类广义超几何函数在z=1点处的值可以通过Karlsson-Minton-Miller型的求和定理来获得.我们首先为这两个分数阶积分算子建立起一系列的基本结果,其中包括:Fox H-函数的广义分数阶积分、映射性质以及它们的Mellin变换.其次,我们研究了这些广义分数阶积分算子的分解结构.更确切地讲,借助于Butzer和Jansche所提出的Mellin变换理论,我们证明了这些算子可以被分解为一系列的Laplace变换L及其逆变换L-1的复合.作为应用,我们推导了两个有关H-函数的非常一般的结果.同时,我们证明了这些分数阶积分算子,当被理解为某种积分方程时,具有L和L-1形式的解.我们还考虑了广义分数阶积分算子的分解结构在某些具体的积分方程中的应用,并发现其中一个方程的解能够由Aleph((?))-函数来表示.接着,我们利用算子I和矩阵分析中积和式的概念,构造了一种新的多重(?)eby(?)ev型泛函,并建立了这类泛函关于同步函数的一些不等式.最后,我们推广了 Erd(?)lyi关于Gauss超几何函数的一个经典积分,使得对于一些特殊的广义超几何函数也有相似形式的积分成立.我们的结论主要依赖于分数阶分部积分和级数操作技巧.我们还得到了一些重要的特殊情形,其中就包括了一类新型的Thomae型变换.
[Abstract]:The main purpose of this paper is to establish a new set of theories for generalized fractional integral operators I and J. The characteristic of these two operators is that their integral kernels contain a very special class of generalized hypergeometric functions r 2Fr 1. At the same time, the importance of operators I and J is also derived from the value of the generalized hypergeometric functions which they contain at point ZG 1, which can be obtained by the sum theorem of Karlsson-Minton-Miller type. We first establish a series of basic results for these two fractional integral operators, including the generalized fractional integral of the: Fox H- function, the mapping properties and their Mellin transformations. Secondly, we study the decomposition structure of these generalized fractional integral operators. More precisely, with the help of the Mellin transformation theory proposed by Butzer and Jansche, we prove that these operators can be decomposed into a series of Laplace transformations L and L ~ (-1). As an application, we derive two very general results about H-functions. At the same time, we prove that these fractional integral operators, when understood as some integral equations, have solutions in the form of L and L ~ (-1). We also consider the application of the decomposition structure of the generalized fractional integral operator to some concrete integral equations, and find that the solution of one of the equations can be represented by the Alephelian function. Then, by using the concept of product summation in operator I and matrix analysis, we construct a new functional of multiplex ebydov type, and establish some inequalities about synchronization functions of this kind of functional. Finally, we generalize a classical integral of Gauss hypergeometric functions by Erd(?)lyi, so that some special generalized hypergeometric functions have similar forms of integrals. Our conclusion mainly depends on fractional partial integral and series operation skill. We also get some important special cases, including a new class of Thomae type transformations.
【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O177.6

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