本文選題：分數(shù)階積分算子 + 分數(shù)階分部積分��； 參考：《華東師范大學》2017年博士論文

【摘要】：本文的主要目的是為廣義分數(shù)階積分算子I和J建立一套新的理論.這兩個算子的特征是其積分核中均包含一類十分特殊的廣義超幾何函數(shù)r+2Fr+1.同時,算子I和J的重要性也正是源于它們所包含的這類廣義超幾何函數(shù)在z=1點處的值可以通過Karlsson-Minton-Miller型的求和定理來獲得.我們首先為這兩個分數(shù)階積分算子建立起一系列的基本結果,其中包括:Fox H-函數(shù)的廣義分數(shù)階積分、映射性質(zhì)以及它們的Mellin變換.其次,我們研究了這些廣義分數(shù)階積分算子的分解結構.更確切地講,借助于Butzer和Jansche所提出的Mellin變換理論,我們證明了這些算子可以被分解為一系列的Laplace變換L及其逆變換L-1的復合.作為應用,我們推導了兩個有關H-函數(shù)的非常一般的結果.同時,我們證明了這些分數(shù)階積分算子,當被理解為某種積分方程時,具有L和L-1形式的解.我們還考慮了廣義分數(shù)階積分算子的分解結構在某些具體的積分方程中的應用,并發(fā)現(xiàn)其中一個方程的解能夠由Aleph((?))-函數(shù)來表示.接著,我們利用算子I和矩陣分析中積和式的概念,構造了一種新的多重(?)eby(?)ev型泛函,并建立了這類泛函關于同步函數(shù)的一些不等式.最后,我們推廣了 Erd(?)lyi關于Gauss超幾何函數(shù)的一個經(jīng)典積分,使得對于一些特殊的廣義超幾何函數(shù)也有相似形式的積分成立.我們的結論主要依賴于分數(shù)階分部積分和級數(shù)操作技巧.我們還得到了一些重要的特殊情形,其中就包括了一類新型的Thomae型變換.
[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.
【學位授予單位】：華東師范大學
【學位級別】：博士
【學位授予年份】：2017
【分類號】：O177.6

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