Kummer’s 24 Solutions of the Hypergeometric Differential Equation with the Aid of Fractional Calculus  

作　　者：Tohru Morita Ken-ichi Sato Tohru Morita;Ken-ichi Sato(Tohoku University, Sendai, Japan;College of Engineering, Nihon University, Koriyama, Japan)

机构地区：[1]Tohoku University, Sendai, Japan [2]College of Engineering, Nihon University, Koriyama, Japan

出　　处：《Advances in Pure Mathematics》2016年第3期180-191,共12页理论数学进展（英文）

摘　　要：We know that the hypergeometric function, which is a solution of the hypergeometric differential equation, is expressed in terms of the Riemann-Liouville fractional derivative (fD). The solution of the differential equation obtained by the Euler method takes the form of an integral, which is confirmed to be expressed in terms of the Riemann-Liouville fD of a function. We can rewrite this derivation such that we obtain the solution in the form of the Riemann-Liouville fD of a function. We present a derivation of Kummer’s 24 solutions of the hypergeometric differential equation by this method.We know that the hypergeometric function, which is a solution of the hypergeometric differential equation, is expressed in terms of the Riemann-Liouville fractional derivative (fD). The solution of the differential equation obtained by the Euler method takes the form of an integral, which is confirmed to be expressed in terms of the Riemann-Liouville fD of a function. We can rewrite this derivation such that we obtain the solution in the form of the Riemann-Liouville fD of a function. We present a derivation of Kummer’s 24 solutions of the hypergeometric differential equation by this method.

关 键 词：Fractional Derivative Hypergeometric Differential Equation Hypergeometric Function 

分 类 号：O17[理学—数学]