Singular integral equation involving a multivariable analog of Mittag-Leffler function
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Singular integral equation involving a multivariable analog of Mittag-Leffler function Sebastien Gaboury1* and Mehmet Ali Özarslan2 * Correspondence: [email protected] 1 Department of Mathematics and Computer Science, University of Quebec at Chicoutimi, Quebec, G7H 2B1, Canada Full list of author information is available at the end of the article
Abstract Motivated by the recent work of the second author (Özarslan in Appl. Math. Comput. 229:350-358, 2014), we present, in this paper, some fractional calculus formulas for a mild generalization of the multivariable Mittag-Leffler function, a Schläfli’s type contour integral representation, some multilinear and mixed multilateral generating functions; and, finally, we consider a singular integral equation with the function (γ ),(1) E(ρrr ),λ (x1 , . . . , xr ) in the kernel and we provide its solution. MSC: 26A33; 33E12 Keywords: fractional integrals and derivatives; Mittag-Leffler function; contour integral representation; generating functions; singular integral equation; Laplace transform
1 Introduction The celebrated Mittag-Leffler function [, ] is defined by
Eα (z) =
∞ k=
zk (αk + )
(.)
α ∈ C; (α) > ; z ∈ C ,
where C denotes the set of complex numbers. The Mittag-Leffler function arises naturally in the solution of fractional integral equations []. A generalization of the Mittag-Leffler function Eα (z) has been investigated by Wiman []. He studied the following function:
Eα,β (z) =
∞ k=
zk (αk + β)
(.)
α, β ∈ C; (α) > ; (β) > ; z ∈ C .
Other generalizations of the Mittag-Leffler functions were given in [, ]. Let us recall the one given by Srivastava and Tomovski []: ©2014 Gaboury and Özarslan; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Gaboury and Özarslan Advances in Difference Equations 2014, 2014:252 http://www.advancesindifferenceequations.com/content/2014/1/252
γ ,K
Eα,β (z) =
∞ k=
(γ )Kn zk (αk + β) k!
Page 2 of 10
(.)
α, β, γ ∈ C; (α) > max , (K) – ; (K) > ; (β) > ; z ∈ C ,
where (λ)κ denotes the Pochhammer symbol defined, in terms of the Gamma function, by ⎧ (λ + κ) ⎨λ(λ + ) · · · (λ + n – ) (κ = n ∈ N; λ ∈ C), = (λ)κ := ⎩ (λ) (κ = ; λ ∈ C \ {}),
(.)
where N denotes the set of positive integers. Multivariable analog of the Mittag-Leffler function has been introduced and investigated by Saxena et al. [, p., Eq. (.)] in the following form: (γ )
(γ ,...,γ )
E(ρrr ),λ (z , . . . , zr ) = E(ρ ,...,ρrr ),λ (z , . . . , zr ) ∞
=
k ,...,kr =
(γ )k · · · (γr )kr zk · · · zrkr (k ρ + · · · + kr ρr + λ) k ! · · · kr !
(.)
λ, zj , γj , ρj ∈ C; (ρj ) > ; j = , , . . . , r .
This function is, in fact, a special case of the generalized Lauricella series in several variables, introduced by Srivastav
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