Asymptotic Expansions of Fractional Derivatives andTheir Applications

Tohru Morita and Ken-ichi Sato
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Tohru Morita: Graduate School of Information Sciences, Tohoku University, Sendai 980-8577, Japan
Ken-ichi Sato: College of Engineering, Nihon University, Koriyama 963-8642, Japan

Mathematics, 2015, vol. 3, issue 2, 1-19

Abstract: We compare the Riemann–Liouville fractional integral (fI) of a function f(z)with the Liouville fI of the same function and show that there are cases in which theasymptotic expansion of the former is obtained from those of the latter and the differenceof the two fIs. When this happens, this fact occurs also for the fractional derivative (fD).This method is applied to the derivation of the asymptotic expansion of the confluenthypergeometric function, which is a solution of Kummer’s differential equation. In thepresent paper, the solutions of the equation in the forms of the Riemann–Liouville fI orfD and the Liouville fI or fD are obtained by using the method, which Nishimoto used insolving the hypergeometric differential equation in terms of the Liouville fD.

Keywords: fractional derivative; asymptotic expansion; Kummer’s differential equation; confluent hypergeometric function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2015
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