Abstract
We aim to present some formulas for the Saigo hypergeometric fractional integral and differential operators involving the generalized Mathieu series S μ ( r ) , which are expressed in terms of the Hadamard product of the generalized Mathieu series S μ ( r ) and the Fox–Wright function p Ψ q ( z ) . Corresponding assertions for the classical Riemann–Liouville and Erdélyi–Kober fractional integral and differential operators are deduced. Further, it is emphasized that the results presented here, which are for a seemingly complicated series, can reveal their involved properties via the series of the two known functions.
Highlights
Introduction and PreliminariesFractional calculus, which has a long history, is an important branch of mathematical analysis where differentiations and integrations can be of arbitrary non-integer order
Several interesting problems and solutions deal with integral representations and bounds for the following mild generalization of the Mathieu series and its alternative version with a fractional power defined by ([10], p. 2, Equation (16))
The results presented in Theorems together with Corollaries are sure to be new and potentially useful, mainly because they are expressed in terms of the Hadamard product with two known functions
Summary
Fractional calculus, which has a long history, is an important branch of mathematical analysis (calculus) where differentiations and integrations can be of arbitrary non-integer order. Several interesting problems and solutions deal with integral representations and bounds for the following mild generalization of the Mathieu series and its alternative version with a fractional power defined by Such a series has been widely considered in mathematical literature (see, e.g., papers of Cerone and Lenard [10], Diananda [12] and Pogány et al [8]). Tomovski and Pogány [14] studied the several integral representations of the generalized fractional order Mathieu-type power series. Our aim is to study the compositions of the generalized fractional integration and differentiation operators (1)–(4) with the generalized Mathieu series (21) in terms of the Hadamard product (22) of the generalized Mathieu series and the Fox–Wright function. A seemingly complicated resulting series expressed in terms of two known functions means that certain properties involved in the complicated resulting series can be revealed via the series of the known functions
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