New Fractional Integrals and Derivatives Results for the Generalized Mathieu-Type and Alternating Mathieu-Type Series

Authored by: Rakesh K. Parmar , Arjun K. Rathie , S. D. Purohit

Handbook of Fractional Calculus for Engineering and Science

Print publication date:  February  2022
Online publication date:  February  2022

Print ISBN: 9781032047799
eBook ISBN: 9781003263517
Adobe ISBN:

10.1201/9781003263517-12

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Abstract

Operators of fractional integration and differentiation have a long history of applications in engineering and sciences. Motivated by the systematic historical account of the investigations of fractional calculus operators results by Srivastava and Saxena [Appl. Math. Comput. 118, (2001), 1–52], we establish some new fractional calculus results for the generalized Mathieu-type and alternating Mathieu-type series by extensive use of Marichev–Saigo–Maeda operator tools. The (presumably) new and (potentially) useful results are expressed in terms of the generalized H-function, that is the Ħ-function [J. Phys. A: Math. Gen. 20 (1987), 4119–4128]. As a special case and from an applications point of view, all the results are also deduced in terms of Fox’s H-function [Trans. Amer. Math. Soc. 98 (1961), 395–429]. We also observe that all the results derived here can also be represented in terms of theI-function introduced in 1997 by Rathie [Le Matematiche 52(2) (1997), 297–310].

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