Mathematical Physics Vol 1

5.7 Mittag-Leffler functions

255

5.7 Mittag-Leffler functions It is known that the solution of a linear differential equation with constant coefficients can be represented as an exponential function. On the other hand, the fractional differential equation with constant coefficients has in many cases a solution given in the form of the so-called Mittag Leffler function . It is not difficult to see, as will be shown later, that the Mittag-Leffler function is a generalization of the exponential function. The function introduced by Mittag-Leffler in 1903, which contains one parameter and which today bears his name, can be considered a generalization of the exponential function, because it is reduced to it when the parameter takes the value one. Mittag-Leffler functions will be presented here as functions of a real argument, as well as of two or three parameters. Definition (Mittag-Leffler function) The Mittag-Leffler function of parameter α , Re ( α ) > 0, also known as the classical Mittag-Leffler function, denoted by E α ( x ) , is defined by E α ( x ) : = ∑ k = 0 x k Γ ( α k + 1 ) . (5.183) E α ( x x α k Γ ( α k + 1 ) . In both cases, for α = 1 the exponential function is obtained, and thus both definitions can be considered fractional generalizations of the exponential function. There are several generalizations of the Mittag-Leffler function, taking the number of parameters as the number of variables. Definition (Two-parameter Mittag-Leffler function) Let x ∈ C , and α , β ∈ C be two parameters, where Re ( α ) > 0 and Re ( β ) > 0. The two-parameter Mittag-Leffler function is defined by E α , β ( x ) : = ∑ k = 0 x k Γ ( α k + β ) . (5.184) α ) : = ∑ k = 0 This two-parameter Mittag-Leffler function is a generalization of the function defined by the previous expression, as for β = 1we obtain E α , 1 ( x )= E α ( x ) . Definition (Three-parameter Mittag-Leffler function) Let x ∈ C and α , β , ρ ∈ C be three parameters, where Re ( α ) > 0, Re ( β ) > 0 and Re ( ρ ) > 0. The three-parameter Mittag-Leffler function is defined as An alternative definitions is

x k k !

( ρ ) k Γ ( α k + β )

E ρ

( x ) : = ∑ k = 0

(5.185)

,

α , β