Fractional Integration and Differentiation of the Generalized Mathieu Series

Abstract

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.

Generalized Fractional Integral Operators Pertaining to the Product of Srivastava’s Polynomials and Generalized Mathieu Series

“…Remark 1. If we set w = 0, A 0,0 = 1 then S u 0 [x] → 1 in Corollaries 1 and 2, we can deduce the known result given by Sexana and Parmar ( [22], Equations (31) and (33)).…”

“…Following the work of Saxena and Parmar [22], our aim is to study the novel combination of the Saigo's fractional integral operators involving the product of Srivastava's polynomials and the generalized Mathieu series. The results are general in nature and expressed in terms of the generalized hypergeometric function and Hadamard product of the generalized Mathieu series.…”

Certain generalized fractional calculus formulas and integral transforms involving $(p,q)$-Mathieu-type series

“…These operators includes Saigo hypergeometric fractional calculus operators, Riemann-Liouville and Erdélyi-Kober fractional calculus operators as special cases for various choices of the parameters (see for details [2,8,10] and [12]). In a recent paper, Saxena and Parmar [9] established several interesting Saigo hypergeometric fractional formulas involving the generalized Mathieu series defined by Tomovski and Pogány [14]. More recently, Singh et al [10] established several results by employing Marichev-Saigo-Maeda fractional operators including their composition formulas and using certain integral transforms involving the extended generalized Mathieu series defined by Tomovski and Mehrez [13].…”

“…In our present investigation, we require the definition of the Hadamard product (or the convolution) of two analytic functions [9]. If the R f and R g are the radii of convergence of the two power series where R = lim n→∞ a n b n a n+1 b n+1 = lim n→∞ a n a n+1 · lim…”

Fractional integral operator associated with extended Mittag-Leffler function

“…It has demonstrated applications in diverse field of applied sciences and mathematics like diffusion, reaction-diffusion, fluid flow, polymer physics, chemical physics etc. A comprehensive account of fractional calculus operators and its applications and Special functions can be found in the monographs written by Agarwal and Choi (2016), Agarwal (2014, 2015), Gehlot (2013), Gupta and Parihar (2017), Kataria and Vellaisamy (2015), Kilbas et al (2004), Nadir and Khan (2018a), Rahman et al (2017aRahman et al ( , 2017b, Rao et al (2010), Saigo (1978), Saigo and Maeda (1998), Samko et al (1993), Saxena and Parmar (2017), Shishkina and Sitnik (2017), Singh (2013), Srivastava and Agarwal (2013), Srivastava et al (2012aSrivastava et al ( , 2012bSrivastava et al ( , 2017, Suthar et al (2017), and references therein.…”