Article
KYUNGPOOK Math. J. 2019; 59(2): 325-333
Published online June 23, 2019
Copyright © Kyungpook Mathematical Journal.
Some Generating Relations of Extended Mittag-Leffler Functions
Nabiullah Khan, Mohd Ghayasuddin, Mohd Shadab∗
Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh-202002, India
e-mail : nukhanmath@gmail.com
Department of Mathematics, Faculty of Science, Integral University, Lucknow226026, India
e-mail : ghayas.maths@gmail.com
Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia(A Central University), New Delhi-110025, India
e-mail : shadabmohd786@gmail.com
Received: March 16, 2017; Revised: March 12, 2019; Accepted: March 18, 2019
Abstract
Motivated by the results on generating functions investigated by H. Exton and many other authors, we derive certain (presumably) new generating functions for generalized Mittag-Leffler-type functions. Specifically, we introduce a new class of generating relations (which are partly bilateral and partly unilateral) involving the generalized Mittag-Leffler function. Also we present some special cases of our main result.
Keywords: generalized Mittag-Leffler’s function, hypergeometric function, generating function.
1. Introduction
In 1903, the Swedish mathematician Gosta Mittag-Leffler [7] introduced the function
where
The Mittag-Leffler function is a direct generalization of the exponential function to which it reduces when
Wiman [17] introduced a new generalization of
which is known as the Wiman function. Properties of the Wiman function
Prabhakar [9] introduced a further generalization of
where
For
The function
In a sequel to the above-mentioned works, Shukla and Prajapati [15] defined the following generalization of the Mittag-Leffler function:
where
In 2009 and 2012, Salim [13] and Salim and Faraj [14] introduced further generalizations of the preceding functions which are given as follows:
and
Subsequently, Khan and Ahmad [6] defined the following two interesting generalizations of these functions and investigated their associated properties:
and
where (
2. Generating Relation
An interesting result on generating functions was given by H. Exton [3, p.147(3)]. The modified form of his result due to Pathan and Yasmeen [8] is
where
Main result
On expanding the function
in series form, we obtain
Replacement of
which is our required result.
3. Special Cases
(1) On setting
where
(2) On setting
where
(3) On setting
where
(4) For
where
(5) On setting
where
(6) On setting
(7) On setting
(8) On setting
(9) On setting
(10) On setting
(11) On setting
4. Concluding Remark
In our present investigation, we have studied a number of generating functions for the extended Mittag-Leffler-type functions given in [4, 6, 13, 14, 15]. The main generating function is the further generalization of the result given by Kamarujjama and Khan [5]. The results of this paper, especially (
Acknowledgements
The authors thank the constructive comments and suggestions by anonymous referees. They have contributed to improve the presentation of this manuscript. The authors wish to acknowledge R. B. Paris (Abertay University, Dundee, UK) for his assistance with the improvement of the text.
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