The extended Mittag-Leffler function and its properties
© Özarslan and Y¿lmaz; licensee Springer. 2014
Received: 5 November 2013
Accepted: 13 December 2013
Published: 20 February 2014
In this paper, we present the extended Mittag-Leffler functions by using the extended Beta functions (Chaudhry et al. in Appl. Math. Comput. 159:589-602, 2004) and obtain some integral representations of them. The Mellin transform of these functions is given in terms of generalized Wright hypergeometric functions. Furthermore, we show that the extended fractional derivative (Özarslan and Özergin in Math. Comput. Model. 52:1825-1833, 2010) of the usual Mittag-Leffler function gives the extended Mittag-Leffler function. Finally, we present some relationships between these functions and the Laguerre polynomials and Whittaker functions.
Fractional differential equations have been an active research area during the past few decades and they occur in many applications of physics and engineering. The Mittag-Leffler function appears as the solution of fractional order differential equations and fractional order integral equations. Some applications of the Mittag-Leffler function are as follows: studies of the kinetic equation, the telegraph equation , random walks, Levy flights, superdiffuse transport, and complex systems. Besides this, the Mittag-Leffler function appears in the solution of certain boundary value problems involving fractional integro-differential equations of Volterra type . It has applications in applied problems, such as fluid flow, rheology, diffusive transport akin to diffusion, electric networks, probability, and statistical distribution theory. Various properties of the Mittag-Leffler functions were presented and surveyed in . Furthermore, a different variant of the Mittag-Leffler function has been investigated in .
where with . For , it reduces to the Mittag-Leffler function given in equation (2). Some of the properties of the generalized Mittag-Leffler function such as the Mellin transform, the inverse Mellin transform, and differentiation were given in . On the other hand, monotony of the Mittag-Leffler function was given in .
The organization of the paper is as follows: In Section 2, we give an integral representations of the extended Mittag-Leffler function in terms of Prabhakar’s Mittag-Leffler function and in terms of known elementary functions. The Mellin transform of the extended Mittag-Leffler function is obtained by means of the generalized Wright hypergeometric function . In Section 3, we obtain fractional derivative representations of the extended Mittag-Leffler function and give some derivative formulas. In Section 4, we obtain the relationship between the extended Mittag-Leffler function and simple Laguerre polynomials and Whittaker’s functions.
2 Some properties of the extended Mittag-Leffler function
We begin with the following theorem, which gives the integral representation of the extended Mittag-Leffler function.
Theorem 1 (Integral representation)
where , , , .
Inserting the above recurrence relation into equation (6), we get the following recurrence relation for the extended Mittag-Leffler’s function.
Corollary 4 (Recurrence relation)
where , , , .
Theorem 5 (Mellin transform)
where is the Wright generalized hypergeometric function.
3 Derivative properties of the extended Mittag-Leffler function
The extended Riemann-Liouville fractional derivative operator was defined by Özarslan and Özergin as follows.
Definition 8 ()
where the path of integration is a line from 0 to z in the complex t-plane. For the case , we obtain the classical Riemann-Liouville fractional derivative operator.
We begin by the following theorem.
Whence the result. □
In the following theorem, we give the derivative properties of the extended Mittag-Leffler function.
Continuing the repetition of this procedure n times, we get the desired result. □
Proof In equation (23), replace z by and multiply , then taking the z-derivative n times, we get the result. □
Proof Taking the p-derivative n times in equation (6), we get the result. □
4 Relations between the extended Mittag-Leffler function with Laguerre polynomial and Whittaker function
In this section, we give a representation of the extended Mittag-Leffler function in terms of Laguerre polynomials and Whittaker’s function.
where , , .
Multiplying both sides of equation (29) by , we get the result. □
In the following theorem, we give the extended Mittag-Leffler function in terms of Whittaker’s function.
in equation (33), we get the result. □
- Camargo RF, Capelas de Oliveira E, Vaz J: On the generalized Mittag-Leffler function and its application in a fractional telegraph equation. Math. Phys. Anal. Geom. 2012,15(1):1-16. 10.1007/s11040-011-9100-8MathSciNetView ArticleMATHGoogle Scholar
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, New York; 1993.MATHGoogle Scholar
- Haubold HJ, Mathai AM, Saxena RK: Mittag-Leffler functions and their applications. J. Appl. Math. 2011., 2011: Article ID 298628 10.1155/2011/298628Google Scholar
- Soubhia AL, Camargo RF, Capelas de Oliveira E, Vaz J: Theorem for series in three-parameter Mittag-Leffler function. Fract. Calc. Appl. Anal. 2010, 13: 1-12.MathSciNetMATHGoogle Scholar
- Mittag-Leffler GM: Une généralisation de l’intégrale de Laplace-Abel. C. R. Acad. Sci., Ser. II 1903, 137: 537-539.MATHGoogle Scholar
- Mittag-Leffler GM:Sur la nouvelle fonction . C. R. Acad. Sci. 1903, 137: 554-558.MATHGoogle Scholar
- Mittag-Leffler GM: Sur la représentation analytique d’une fonction monogéne (cinquiéme note). Acta Math. 1905,29(1):101-181. 10.1007/BF02403200MathSciNetView ArticleMATHGoogle Scholar
- Wiman A:Über der Fundamentalsatz in der Theorie der Funktionen . Acta Math. 1905, 29: 191-201. 10.1007/BF02403202MathSciNetView ArticleMATHGoogle Scholar
- Wiman A:Über die Nullstellen der Funktionen . Acta Math. 1905, 29: 217-234. 10.1007/BF02403204MathSciNetView ArticleMATHGoogle Scholar
- Agarwal RP: A propos d’une note de M. Pierre Humbert. C. R. Séances Acad. Sci. 1953, 236: 2031-2032.MathSciNetMATHGoogle Scholar
- Humbert P, Agarwal RP: Sur la fonction de Mittag-Leffler et quelques unes de ses generalizations. Bull. Sci. Math., Ser. II 1953, 77: 180-185.MathSciNetMATHGoogle Scholar
- Prabhakar TR: A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 1971, 19: 7-15.MathSciNetMATHGoogle Scholar
- Kurulay M, Bayram M: Some properties of the Mittag-Leffler functions and their relation with the Wright function. Adv. Differ. Equ. 2012., 2012: Article ID 178Google Scholar
- Mainardi, F: On some properties of the Mittag-Leffler function , completely monotone for t > 0 with 0 < α < 1. arXiv: http://arxiv.org/abs/arXiv:1305.0161
- Chaudhry MA, Qadir A, Srivastava HM, Paris RB: Extended hypergeometric and confluent hypergeometric functions. Appl. Math. Comput. 2004, 159: 589-602. 10.1016/j.amc.2003.09.017MathSciNetView ArticleMATHGoogle Scholar
- Chaudhry MA, Zubair SM: On a Class of Incomplete Gamma Functions with Applications. 2002.MATHGoogle Scholar
- Srivastava HM, Manocha HL: A Treatise on Generating Functions. Halsted, New York; 1984. (Ellis Horwood, Chichester)MATHGoogle Scholar
- Özarslan MA, Özergin E: Some generating relations for extended hypergeometric functions via generalized fractional derivative operator. Math. Comput. Model. 2010, 52: 1825-1833. 10.1016/j.mcm.2010.07.011View ArticleMathSciNetMATHGoogle Scholar
- Özarslan MA: Some remarks on extended hypergeometric, extended confluent hypergeometric and extended Appell’s functions. J. Comput. Anal. Appl. 2012,14(6):1148-1153.MathSciNetMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. 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 credited.