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'''Prabhakar function''' is a certain special function in mathematics introduced by the Indian mathematician Tilak Raj Prabhakar in a paper published in 1971.<ref name="Tilak">{{cite journal |last1=Tilak Raj Prabhakar |title=A singular integral equation with a generalized Mittag-Leffler function in the kernel |journal=Yoklohama Mathematics Journal |date=1971 |volume=19 |issue=1 |pages=7–15 |url=https://ynu.repo.nii.ac.jp/record/6514/files/YMJ_19_N1_1971_007-015.pdf |access-date=27 December 2023}}</ref> 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 equation|integral equation]]s. Later the function was found to have applications in the theory of [[Fractional calculus|fractional calculus]] and also in certain areas of physics.<ref name="Andrea"/>
==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, <math>\Gamma (z)</math> is the well known [[Gamma function|gamma function]] defined by 

::<math>\Gamma(z)= \int_0^\infty t^{z-1}e^{-z}\, dz, \quad\Re(z) > 0</math>.

In the following it will be assumed that <math>\alpha</math>, <math>\beta</math> and <math>\gamma</math> are all complex numbers. 

=== One-parameter Mittag-Leffler function ===
The one-parameter Mittag-Leffler function is defined as<ref>{{cite book |last1=Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. Rogosin |title=Mittag-Leffler Functions, Related Topics and Applications |date=2014 |publisher=Springer |isbn=978-3-662-43929-6|page=17}}</ref> 

::<math>E_\alpha(z)=\sum_{n=0}^\infty \dfrac{z^n}{\Gamma(\alpha n + 1)}.</math>

=== Two-parameter Mittag-Leffler function ===
The two-parameter Mittag-Leffler function is defined as<ref>{{cite book |last1=Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. Rogosin |title=Mittag-Leffler Functions, Related Topics and Applications |date=2014 |publisher=Springer |isbn=978-3-662-43929-6|page=56}}</ref> 

::<math>E_{\alpha, \beta}(z)=\sum_{n=0}^\infty \dfrac{z^n}{\Gamma(\alpha n + \beta)}, \quad \Re(\alpha) >0.</math>

=== Three-parameter Mittag-Leffler function (Prabhakar function) ===
The three-parameter Mittag-Leffler function (Prabhakar function) is defined by<ref name="Tilak"/><ref>{{cite book |last1=Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. Rogosin |title=Mittag-Leffler Functions, Related Topics and Applications |date=2014 |publisher=Springer |isbn=978-3-662-43929-6|page=97}}</ref><ref>{{cite journal |last1=Roberto Garra and Roberto Garrappa |title=The Prabhakar or three parameter Mittag–Leffler function: theory and application |journal=Communications in Nonlinear Science and Numerical Simulation |date=2018 |volume=56 |pages=314–329 |doi=10.1016/j.cnsns.2017.08.018 |arxiv=1708.07298v2 |bibcode=2018CNSNS..56..314G }}</ref>

::<math>E_{\alpha, \beta}^\gamma (z)=\sum_{n=0}^\infty \dfrac{(\gamma)_n}{n!\Gamma(\alpha n + \beta)} z^n, \quad \Re(\alpha)>0</math>

where <math>(\gamma)_n=\gamma(\gamma+1)\ldots (\gamma+n-1)</math>.

==Elementary special cases==
The following special cases immediately follow from the definition.<ref name="Andrea">{{cite journal |last1=Andrea Giusti, Ivano Colombaro, Roberto Garra |title=A practical guide to Prabhakar fractional calculus |journal=Fractional Calculus and Applied Analysis |date=2020 |volume=25 |issue=1 |pages=9–54 |doi=10.1515/fca-2020-0002 |arxiv=2002.10978 }}</ref>

# <math>E_{\alpha,\beta}^0 (z)= \frac{1}{\Gamma(\beta)}</math>
# <math>E_{\alpha,\beta}^1 (z)= E_{\alpha,\beta}(z)</math>, the two-parameter Mittag-Leffler function.
# <math>E_{\alpha,1}^1 (z)= E_{\alpha}(z)</math>, the one-parameter Mittag-Leffler function.
# <math>E_{1, 1}^1 (z) = e^z</math>, the classical exponential function.

==Properties==

=== Reduction formula ===

The following formula can be reduced to lower the value of the third parameter <math>\gamma</math>.<ref name="Andrea"/>

:: <math> 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]</math>

=== Relation with Fox–Wright function ===

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

::<math>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)</math>

=== Derivatives ===

The derivative of the Prabhakar function is given by

::<math> \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] </math>

There is a general expression for higher order derivatives. Let <math>m</math> be a positive integer. The <math>m</math>-th derivative of the Prabhakar function is given by

::<math> \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)</math>

The following result is useful in applications.

::<math>\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)</math>

=== Integrals ===

The following result involving Prabhakar function is known.

::<math> \int_0^t  \tau^{\beta-1} E_{\alpha,\beta}^\gamma(\tau^\alpha z) = t^\beta E_{\alpha, \beta+1}^\gamma(t^\alpha z)</math>

=== Laplace transforms ===

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

::<math>  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}</math>

==Prabhakar fractional calculus==

The following function is known as the Prabhakar kernel in the literature.<ref name="Andrea"/>

::<math> e_{\alpha,\beta}^\gamma (t;\lambda) = t^{\beta -1} E_{\alpha,\beta}^\gamma(t^\alpha z)</math>

Given any function <math>f(t)</math>, the convolution of the Prabhakar kernel and <math>f(t)</math> is called the Prabhakar fractional integral:

::<math> \int_{t_0}^t (t-u)^{\beta -1} E_{\alpha,\beta}^\gamma \left( \lambda (t-u)^\alpha\right) f(u)\, du</math>

Properties of the Prabhakar fractional integral have been extensively studied in the literature.<ref>{{cite journal |last1=Anatoly A. Kilbas, Megumi Saigo and R. K. Saxena |title=Generalized mittag-leffler function and generalized fractional calculus operators |journal=Integral Transforms and Special Functions |date=2004 |volume=15 |issue=1 |pages=31–49|doi=10.1080/10652460310001600717 |s2cid=120569191 }}</ref><ref>{{cite journal |last1=F. Polito and Z. Tomovski |title=Some properties of Prabhakar-type fractional calculus operators |journal=Fractional Differential Calculus |date=2016 |volume=6 |issue=1 |pages=73–94|doi=10.7153/fdc-06-05 |arxiv=1508.03224 }}</ref>

==References==
{{reflist}}


[[Category:Special functions]]
[[Category:Fractional calculus]]
[[Category:Integral equations]]

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