Prabhakar function

One-parameter Mittag-Leffler function

The one-parameter Mittag-Leffler function is defined as

${\displaystyle E_{\alpha }(z)=\sum _{n=0}^{\infty }{\dfrac {z^{n}}{\Gamma (\alpha n+1)}}.}$

Two-parameter Mittag-Leffler function

The two-parameter Mittag-Leffler function is defined as

${\displaystyle E_{\alpha ,\beta }(z)=\sum _{n=0}^{\infty }{\dfrac {z^{n}}{\Gamma (\alpha n+\beta )}},\quad \Re (\alpha )>0.}$

Three-parameter Mittag-Leffler function (Prabhakar function)

The three-parameter Mittag-Leffler function (Prabhakar function) is defined by

${\displaystyle E_{\alpha ,\beta }^{\gamma }(z)=\sum _{n=0}^{\infty }{\dfrac {(\gamma )_{n}}{n!\Gamma (\alpha n+\beta )}}z^{n},\quad \Re (\alpha )>0}$

where ${\displaystyle (\gamma )_{n}=\gamma (\gamma +1)\ldots (\gamma +n-1)}$.

Reduction formula

The following formula can be reduced to lower the value of the third parameter ${\displaystyle \gamma }$.

${\displaystyle E_{\alpha ,\beta }^{\gamma +1}(z)={\frac {1}{\alpha \gamma }}{\big [}E_{\alpha ,\beta -1}^{\gamma }(z)+(1-\beta +\alpha \gamma )E_{\alpha ,\beta }^{\gamma }(z){\big ]}}$

Relation with Fox–Wright function

The Prabhakar function is related to the Fox–Wright function by the following relation:

${\displaystyle E_{\alpha ,\beta }^{\gamma }(z)={\frac {1}{\Gamma (\gamma )}}{}_{1}\Psi _{1}\left({\begin{matrix}\left(\gamma ,1\right)\\(\beta ,\alpha )\end{matrix}};z\right)}$

Derivatives

The derivative of the Prabhakar function is given by

${\displaystyle {\frac {d}{dz}}\left(E_{\alpha ,\beta }^{\gamma }(z)\right)={\frac {1}{\alpha z}}{\big [}E_{\alpha ,\beta -1}^{\gamma }(z)+(1-\beta )E_{\alpha ,\beta }^{\gamma }{\big ]}}$

There is a general expression for higher order derivatives. Let ${\displaystyle m}$ be a positive integer. The ${\displaystyle m}$-th derivative of the Prabhakar function is given by

${\displaystyle {\frac {d^{m}}{dz^{m}}}\left(E_{\alpha ,\beta }^{\gamma }(z)\right)={\frac {\Gamma (\gamma +m)}{\Gamma (\gamma )}}E_{\alpha ,m\alpha +\beta }^{\gamma +m}(z)}$

The following result is useful in applications.

${\displaystyle {\frac {d^{m}}{dz^{m}}}\left(t^{\beta -1}E_{\alpha ,\beta }^{\gamma }(t^{\alpha }z)\right)=t^{\beta -m-1}E_{\alpha ,\beta -m}^{\gamma }(t^{\alpha }z)}$

Integrals

The following result involving Prabhakar function is known.

${\displaystyle \int _{0}^{t}\tau ^{\beta -1}E_{\alpha ,\beta }^{\gamma }(\tau ^{\alpha }z)=t^{\beta }E_{\alpha ,\beta +1}^{\gamma }(t^{\alpha }z)}$

Laplace transforms

The following result involving Laplace transforms plays an important role in both physical applications and numerical computations of the Prabhakar function.

${\displaystyle L\left[t^{\beta -1}E_{\alpha ,\beta }^{\gamma }(t^{\alpha }z)\,;\,s\right]={\frac {s^{\alpha \gamma -\beta }}{(s^{\alpha }-z)^{\gamma }}},\quad \Re (s)>0,\quad |s|>|z|^{1/\alpha }}$