The Mittag-Leffler function arises naturally in the solution of fractional integral equations (Saxena et al. 2002), and especially in the study of
the fractional generalization of the kinetic equation, random
walks, Lévy flights, and so-called superdiffusive
transport. The ordinary and generalized Mittag-Leffler functions interpolate between
a purely exponential law and power-like behavior of phenomena governed by ordinary
kinetic equations and their fractional counterparts (Lang 1999ab, Hilfer 2000, Saxena
et al. 2002).
Special values for integer are
(4)
(5)
(6)
(7)
(8)
For half-integers , the functions can be written explicitly as
(9)
giving the special value
(10)
(11)
for ,
where
is erf and is erfc (Saxena et al. 2002).
As can be seen, is closely related to Dawson's
integral .
The more general Mittag-Leffler function
(12)
can also be defined for (Wiman 1905, Agarwal 1953, Humbert 1953, Humbert
and Agarwal 1953, Gorenflo 1987, Miller 1993, Mainardi and Gorenflo 1996, Gorenflo
1998, Sixdeniers et al. 1999), so that
(13)
The general Mittag-Leffler function can be representation in terms of Fox
H-functions (Saxena et al. 2002).
The general Mittag-Leffler function satisfies
(14)
for
(Erdélyi et al. 1981, p. 210; Samko et al. 1993, p. 21),
which gives the Laplace transform of as
Agarwal, R. P. "A propos d'une note de M. Pierre Humbert." Comptes Rendus Acad. Sci. Paris236, 2031-2032, 1953.Dzherbashyan,
M. M. Integral Transform Representations of Functions in the Complex Domain.
Moscow: Nauka, 1966.Erdélyi, A.; Magnus, W.; Oberhettinger, F.;
and Tricomi, F. G. "Mittag-Leffler's Function and Related Functions." §18.1 in Higher
Transcendental Functions, Vol. 3. New York: Krieger, pp. 206-212,
1981.Gorenflo, R. "Newtonsche Aufheizung, Abelsche Integralgleichungen
zweiter Art und Mittag-Leffler-Funktionen." Z. Naturforsch. A42,
1141-1146, 1987.Gorenflo, R.; Kilbas, A. A.; and Rogosin, S. V.
"On the Generalized Mittag-Leffler Type Functions." Integral Transform.
Spec. Funct.7, 215-224, 1998.Hilfer, R. and Anton, L. "Fractional
Master Equations and Fractal Time Random Walks." Phys. Rev. E51,
R848-R851, 1995.Hilfer, R. "On Fractional Diffusion and Its Relation
with Continuous Time Random Walks." In Anomalous
Diffusion: From Basics to Application: Proceedings of the XIth Max Born Symposium
Held at Ladek Zdroj, Poland, 20-27 May 1998 (Ed. R. Kutner, A. Pekalski,
and K. Sznaij-Weron). Berlin: Springer-Verlag, pp. 77-82, 1999.Hilfer,
R. (Ed.). Applications
of Fractional Calculus in Physics. Singapore: World Scientific, 2000.Humbert,
P. "Quelques résultats relatifs à la fonction de Mittag-Leffler."
Comptes Rendus Acad. Sci. Paris236, 1467-1468, 1953.Humbert,
P. and Agarwal, R. P. "Sur la fonction de Mittag-Leffler et quelques-unes
de ses généralisations." Bull. Sci. Math. Ser. 277,
180-185, 1953.Humbert, P. and Delerue, P. "Sur une extension à
deux variables de la fonction de Mittag-Leffler." Comptes Rendus Acad. Sci.
Paris237, 1059-1060, 1953.Lang, K. R. Astrophysical
Formulae, Vol. 1: Radiation, Gas Processes, and High-Energy Astrophysics, 3rd
enl. rev. ed. New York: Springer-Verlag, 1999a.Lang, K. R.
Astrophysical Formulae, Vol. 2: Space, Time, Matter and Cosmology. New
York: Springer-Verlag, 1999b.Mainardi, F. and Gorenflo, R. "The
Mittag-Leffler Function in the Riemann-Liouville Fractional Calculus." In Proceedings
of the International Conference Dedicated to the Memory of Academician F. D. Gakhov;
Held in Minsk, February 16-20, 1996 (Ed. A. A. Kilbas). Minsk, Beloruss: Beloruss.
Gos. Univ., Minsk, pp. 215-225, 1996.Meerschaert, M. M.; Benson,
D. A.; Scheffler, H.-P.; and Baeumer, B. "Stochastic Solution of Space-Time
Fractional Diffusion Equations." Phys. Rev. E65, 041103, 2002.Miller,
K. S. "The Mittag-Leffler and Related Functions." Integral Transform.
Spec. Funct.1, 41-49, 1993.Mittag-Leffler, M. G. "Sur
la nouvelle fonction ." Comptes Rendus Acad. Sci. Paris137,
554-558, 1903.Mittag-Leffler, M. G. "Sur la representation
analytique d'une branche uniforme d'une fonction monogene." Acta Math.29,
101-181, 1905.Muldoon, M. E. and Ungar, A. A. "Beyond
Sin and Cos." Math. Mag.69, 3-14, 1996.Podlubny,
I. "The Laplace Transform Method for Linear Differential equations of the Fractional
Order." 30 Oct 1997. http://arxiv.org/abs/funct-an/9710005.Samko,
S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional
Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, pp. 21-22,
1993.Saxena, R. K.; Mathai, A. M.; and Haubold, H. J.
"On Fractional Kinetic Equations." 23 Jun 2002. http://arxiv.org/abs/math.CA/0206240.Sixdeniers,
J.-M.; Penson, K. A.; and Solomon, A. I. "Mittag-Leffler Coherent
States." J. Phys. A: Math. Gen.32, 7543-7563, 1999.Sokolov,
I. M.; Klafter, J. and Blumen, A. "Do Strange Kinetics Imply Unusual Thermodynamics?"
Phys. Rev. E64, 021107, 2001.Wiman, A. "Über
den Fundamentalsatz in der Theorie der Funktionen ." Acta Math.29, 191-201, 1905.