Some properties of the Mittag-Leffler functions and their relation with the Wright functions
- Muhammet Kurulay1, 2Email author and
- Mustafa Bayram2
https://doi.org/10.1186/1687-1847-2012-181
© Kurulay and Bayram; licensee Springer 2012
Received: 6 August 2012
Accepted: 4 October 2012
Published: 17 October 2012
Abstract
This paper is a short description of our recent results on an important class of the so-called Mittag-Leffler functions, which became important as solutions of fractional order differential and integral equations, control systems and refined mathematical models of various physical, chemical, economical, management and bioengineering phenomena. We have studied the Mittag-Leffler functions as their typical representatives, including many interesting special cases that have already proven their usefulness in fractional calculus and its applications. We obtained a number of useful relationships between the Mittag-Leffler functions and the Wright functions. The Wright function plays an important role in the solution of a linear partial differential equation. The Wright function, which we denote by , is so named in honor of Wright who introduced and investigated this function in a series of notes starting from 1933 in the framework of the asymptotic theory of partitions.
MSC:33E12.
Keywords
1 Introduction
1.1 The Mittag-Leffler function
Some new properties of the Mittag-Leffler function, including a definite integral and recurrence relation, were investigated in [6, 7].
1.2 The Wright function
The Wright function plays an important role in the solution of a linear partial differential equation. The Wright function, which we denote by , is so named in honor of Wright, who introduced and investigated this function in a series of notes starting from 1933 in the framework of the asymptotic theory of partitions. This function was introduced that related Mittag-Leffler [8–11]. We obtained a number of useful relationships between the Mittag-Leffler functions and the Wright functions.
1.2.1 Definition
1.2.2 The integral representation of the Wright function
1.2.3 The Laplace transform of the Wright function
2 Some properties of the Mittag-Leffler functions
Theorem 1 (Derivative of the Mittag-Leffler function)
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Theorem 2 (Integration of the Mittag-Leffler function)
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Declarations
Acknowledgements
This work was supported by the scientific and technological research council of Turkey (TUBITAK).
Authors’ Affiliations
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