Introduction: Mittag-Leffler and Other Related Functions

Abstract

The analysis of fractional differential equations, carried out by means of fractional calculus and integral transforms (Laplace, Fourier), leads to certain special functions of Mittag-Leffler (M-L) and Wright types. These useful special functions are investigated systematically as relevant cases of the general class of functions which are popularly known as Fox H-functions, named after Charles Fox, who initiated a detailed study of these functions as symmetrical Fourier kernels (Fox, Trans Am Math Soc 98:395, 1961). Definitions, some properties, relations, asymptotic expansions, and Laplace transform formulas for the M-L type functions and Fox H-function are given in this chapter. At the beginning of the twentieth century, Swedish mathematician Gosta Mittag-Leffler introduced a generalization of the exponential function, today known as the M-L function (Mittag-Leffler, C R Acad Sci Paris 137:554, 1903). The properties of the M-L function and its generalizations had been totally ignored by the scientific community for a long time due to their unknown application in the science. In 1930 Hille and Tamarkin solved the Abel-Volterra integral equation in terms of the M-L function (Hille and Tamarkin, Ann Math 31:479, 1930). The basic properties and relations of the M-L function appeared in the third volume of the Bateman project in the Chapter XVIII: Miscellaneous Functions (Erdelyi et al., Higher Transcendental Functions, vol. 3. McGraw-Hill, New York, 1955). More detailed analysis of the M-L function and their generalizations as well as the fractional derivatives and integrals were published later, and it has been shown that they are of great interest for modeling anomalous diffusion and relaxation processes. Similarly, Fox H-function, introduced by Fox (Trans Am Math Soc 98:395, 1961), is of great importance in solving fractional differential equations and for analysis of anomalous diffusion processes. The Fox H-function has been used to express the fundamental solution of the fractional diffusion equation obtained from a continuous time random walk model. Therefore, in this chapter we will give the most important definitions, relations, asymptotic expansions of these functions which represent a basis for investigation of anomalous diffusion and non-exponential relaxation in different complex systems.