Prabhakar function is a certain special function in mathematics introduced by the Indian mathematician Tilak Raj Prabhakar in a paper published in 1971. The function is a three-parameter generalization of the well known two-parameter Mittag-Leffler function in mathematics. The function was originally introduced to solve certain classes of integral equations. Later the function was found to have applications in the theory of fractional calculus and also in certain areas of physics.
Definition The one-parameter and two-parameter Mittag-Leffler functions are defined first. Then the definition of the three-parameter Mittag-Leffler function, the Prabhakar function, is presented. In the following definitions, Γ ( z ) {\displaystyle \Gamma (z)} is the well known gamma function defined by
Γ ( z ) = ∫ 0 ∞ t z − 1 e − z d z , ℜ ( z ) > 0 {\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-z}\,dz,\quad \Re (z)>0} . In the following it will be assumed that α {\displaystyle \alpha } , β {\displaystyle \beta } and γ {\displaystyle \gamma } are all complex numbers.
One-parameter Mittag-Leffler function The one-parameter Mittag-Leffler function is defined as
E α ( z ) = ∑ n = 0 ∞ z n Γ ( α n + 1 ) . {\displaystyle E_{\alpha }(z)=\sum _{n=0}^{\infty }{\dfrac {z^{n}}{\Gamma (\alpha n+1)}}.}
Two-parameter Mittag-Leffler function The two-parameter Mittag-Leffler function is defined as
E α , β ( z ) = ∑ n = 0 ∞ z n Γ ( α n + β ) , ℜ ( α ) > 0. {\displaystyle E_{\alpha ,\beta }(z)=\sum _{n=0}^{\infty }{\dfrac {z^{n}}{\Gamma (\alpha n+\beta )}},\quad \Re (\alpha )>0.}
Three-parameter Mittag-Leffler function (Prabhakar function) The three-parameter Mittag-Leffler function (Prabhakar function) is defined by
E α , β γ ( z ) = ∑ n = 0 ∞ ( γ ) n n ! Γ ( α n + β ) z n , ℜ ( α ) > 0 {\displaystyle E_{\alpha ,\beta }^{\gamma }(z)=\sum _{n=0}^{\infty }{\dfrac {(\gamma )_{n}}{n!\Gamma (\alpha n+\beta )}}z^{n},\quad \Re (\alpha )>0}
where ( γ ) n = γ ( γ + 1 ) … ( γ + n − 1 ) {\displaystyle (\gamma )_{n}=\gamma (\gamma +1)\ldots (\gamma +n-1)} .
Elementary special cases The following special cases immediately follow from the definition.
E α , β 0 ( z ) = 1 Γ ( β ) {\displaystyle E_{\alpha ,\beta }^{0}(z)={\frac {1}{\Gamma (\beta )}}}
E α , β 1 ( z ) = E α , β ( z ) {\displaystyle E_{\alpha ,\beta }^{1}(z)=E_{\alpha ,\beta }(z)} , the two-parameter Mittag-Leffler function.
E α , 1 1 ( z ) = E α ( z ) {\displaystyle E_{\alpha ,1}^{1}(z)=E_{\alpha }(z)} , the one-parameter Mittag-Leffler function.
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