Certain fractional calculus formulas involving extended generalized Mathieu series

Abstract

We establish fractional integral and derivative formulas by using fractional calculus operators involving the extended generalized Mathieu series. Next, we develop their composition formulas by applying the integral transforms. Finally, we discuss special cases.

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

“…Furthermore, all the corollaries obtained earlier by Singh et al [10] in the same paper can also be written correctly by our results. For example, the corrected version of the first result of Singh et al [10] as given in Theorem 1 should read as follows.…”

“…We left these as an exercise for the interested reader. Furthermore, if we take p = q = 0, λ 1 = λ, λ 2 = λ 3 , = ρ, σ 1 = σ and σ 1 = σ , we recover the 16 known results in corrected form recorded in [10]. Furthermore, all the corollaries obtained earlier by Singh et al [10] in the same paper can also be written correctly by our results.…”

“…Furthermore, if we take p = q = 0, λ 1 = λ, λ 2 = λ 3 , = ρ, σ 1 = σ and σ 1 = σ , we recover the 16 known results in corrected form recorded in [10]. Furthermore, all the corollaries obtained earlier by Singh et al [10] in the same paper can also be written correctly by our results. For example, the corrected version of the first result of Singh et al [10] as given in Theorem 1 should read as follows.…”

“…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].…”

“…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].…”