Inversion of integral transform with an extended generalized Mittag-Leffler function

Mittag-Leffler Functions, Related Topics and Applications

“…It defines a new function which is called the extended generalized Mittag-Leffler function and denoted E˛1 ;ˇ1I˛2;ˇ2 .z/ (see [KilKor05,KilKor06a]). with a curve starting at C1 C i' 1 and ending at C1 C i' 2 ( 1 < ' 1 < ' 2 < C1), leaving the poles of .s/ at the left and the poles of .1 s/ at the right.…”

“…[KilKor06a,KilKor06b]). it is the fourparametric generalized Mittag-Leffler function defined by (6.1.1)), and for˛1C˛2 < 0 the kernel E˛1 ;ˇ1I˛2;ˇ2 is the extended four-parametric generalized Mittag-Leffler function defined by (6.1.22).…”

“…Analogues of the Cauchy-Hadamard, Abel, Tauber and Hardy-Littlewood theorems for the three multi-index Mittag-Leffler functions have also been presented.The extension of the Mittag-Leffler function to a wider set of parameters by using Mellin-Barnes integrals was realized in a series of papers[KilKor05,KilKor06a,KilKor06b,KilKor06c] (see also the paper). / k is the Pochhammer symbol,˛j ;ˇj ; j 2 C; j D 1; : : : ; m, Re˛j > 0.…”

Multi-parametric mittag-leffler functions and their extension

“…The central idea is to change the contour of integration in the Mellin-Barnes integral representation, in order to define corresponding integrals for another range of parameters. We follow here the details and results in the series of articles bu Kilbas and Koroleva [21]- [25].…”

“…on a space of Lebesgue measurable functions are obtained in [24], and its inversion formulas are established in [25]. We note that recently the functions E α,β (z) and E α 1 ,β 1 ;α 2 ,β 2 (z) in (3.8) and (3.9) and their various generalizations and modifications have attracted much attention due to their application to the solution of the integral and differential equations of fractional order [21], [30,.…”