Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

256

where ( ρ ) k denotes the so-called Pochhammer symbol definedby

Γ ( ρ + n ) Γ ( ρ )

( ρ ) n = ρ ( ρ + 1 ) ··· ( ρ + n − 1 )=

, n ≥ 0 , ( ρ ) 0 = 1 .

(5.186)

In the case when ρ = 1 it follows that E 1 α , β ( x )= E α , β ( x ) . Shukla and Prajapati [ 68 ] introduce the Mittag-Leffler function with four parameters defined as follows:

Definition (Mittag-Leffler function with four parameters) Mittag-Leffler function with four parameters is defined by the series

x k k !

( ρ ) qk Γ ( k α b ) k !

E ρ , q α , β

( z ) : = ∑ k = 0

(5.187)

,

Γ ( ρ + qk ) Γ ( ρ ) aˇre four parameters.

where Re ( α ) > 0, Re ( ρ ) > 0 and ( ρ ) qk =

z , Mittag-Leffler function E

Generally speaking, the exponential function e function E α , β ( z ) and Prabhakar function E ρ α , β

α ( z ) ,Wiman

R

( z ) are special cases of Shukla function

E ρ , q α , β

( z ) .

Relation between two Mittag-Leffler functions Let x ∈ C and α , β ∈ C be two parameters, where Re ( α ) > 0, Re ( β ) > 0 and k = 1 , 2 ,... . The following relation between two Mittag-Leffler functions is true d k d x k E α , β ( x )= k ! E k + 1 α , β + α k ( x ) . (5.188) We shall now introduce two other functions that appear in different practical problems, related to Mittag-Leffler functions.

Definition (Miller-Ross function)

The function

∞ ∑ k = 0

( ax ) k Γ ( ν + k + 1 )

E x ( ν , α )= x ν

(5.189)

where x ∈ R , Re ( ν ) > − 1 and a ∈ R , is called the Miller-Ross function .

Definition (Rabotnov function)

The function

∞ ∑ k = 0

( β k x ) k ( α + 1 ) Γ ( 1 + α )( k + 1 )

R α ( β , x )= x α

(5.190)

where Re ( α ) > − 1 i β ∈ R , is called the Rabotnov function .