TITLE:
Kummer’s 24 Solutions of the Hypergeometric Differential Equation with the Aid of Fractional Calculus
AUTHORS:
Tohru Morita, Ken-ichi Sato
KEYWORDS:
Fractional Derivative, Hypergeometric Differential Equation, Hypergeometric Function
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.6 No.3,
February
29,
2016
ABSTRACT: 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.