Mittag-Leffler Function

The Mittag-Leffler function (Mittag-Leffler 1903, 1905) is an entire function defined by the series

for . It is related to the generalized hyperbolic functions by

and given explicitly in terms of generalized confluent hypergeometric functions as

It is implemented in the Wolfram Language as MittagLefflerE[a, z] and MittagLefflerE[a, b, z].

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

For half-integers , the functions can be written explicitly as

giving the special value

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

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

The general Mittag-Leffler function can be representation in terms of Fox H-functions (Saxena et al. 2002).

The general Mittag-Leffler function satisfies

for (Erdélyi et al. 1981, p. 210; Samko et al. 1993, p. 21), which gives the Laplace transform of as

for and (Samko 1993, p. 21; Saxena et al. 2002).