Singular integral equation involving a multivariable analog of Mittag-Leffler function
© Gaboury and Özarslan; licensee Springer. 2014
Received: 10 July 2014
Accepted: 3 September 2014
Published: 24 September 2014
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 in the kernel and we provide its solution.
where ℂ denotes the set of complex numbers.
where ℕ denotes the set of positive integers.
Note that by further specializing the several parameters involved, we can obtain many well-known classes of polynomials such as the Laguerre polynomials of r variables defined by Erdélyi  and the Konhauser polynomials .
Obviously, setting () leads to (1.8).
In this paper, we obtain a Schläfli’s type contour integral representation for the multivariable polynomials given in (1.9). Next, we give some multilinear and mixed multilateral generating functions. We also recall the fractional order integral of the generalized multivariable Mittag-Leffler function. Finally, we consider a singular integral equation with in the kernel and we give its solution. Throughout this paper, the variables are assumed to be real variables.
2 Schläfli’s type contour integral representation of
The Schläfli’s type contour integral representation of in terms of is given in the next theorem.
we find from (2.3) and (2.4) the result asserted by Theorem 2.1. □
3 Multilinear and multilateral generating functions
We begin this section by proving a linear generating function for the polynomials by means of the mild generalization of the multivariate analog of Mittag-Leffler functions.
where we have interchanged the order of summations which is guaranteed because of the uniform convergence of the series under the conditions (). □
Now let , , , , , be complex j-tuples. By making use of the above theorem we have the following.
provided that each member of (3.3) exists and ().
Proof Following similar lines to , the proof is completed. □
4 Fractional integrals and derivatives
In this section, we first recall the definitions of the Riemann-Liouville fractional integrals and derivatives. Next, we give the fractional integral and derivative of the generalized multivariable Mittag-Leffler function where are real variables for .
where denotes the integral part of .
Further special cases of (4.5) and (4.6) can be obtained by suitably specializing the coefficients involved. For instance, if we set (), then (4.5) and (4.6) reduce to two results obtained by Saxena et al. .
We end this section by giving a recurrence relation for the generalized multivariable Mittag-Leffler function .
5 Singular integral equation
In this section, we solve a singular integral equation with the generalized multivariable Mittag-Leffler function in the kernel. To do so, we first find the Laplace transform of the function and we compute an integral involving the product of two generalized multivariable Mittag-Leffler functions.
Finally, taking the inverse Laplace transform on both sides of (5.8), the result follows. □
provided that exists for and is locally integrable for .
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