Prabhakar function explained

Prabhakar function is a certain special function in mathematics introduced by the Indian mathematician Tilak Raj Prabhakar in a paper published in 1971.^[1] 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,

\Gamma (z) is the well known gamma function defined by \Gamma(z) = \int ^{inf ty} _{0} t ^{z-1} e ^{-z} dz, \Re(z) > 0 .

In the following it will be assumed that

\alpha , \beta and \gamma are all complex numbers.

One-parameter Mittag-Leffler function

The one-parameter Mittag-Leffler function is defined as^[2]

E _{\alpha(z)=\sum} ^{inf ty} ^{n =0} \dfrac{z ^{n}{\Gamma(\alpha} n + 1 )}.

Two-parameter Mittag-Leffler function

The two-parameter Mittag-Leffler function is defined as^[3]

E _{\alpha,} (z) = \sum ^{inf ty} _{n =0} \dfrac{z ^{n}{\Gamma(\alpha} n + \beta)}, \Re(\alpha) > 0 .

Three-parameter Mittag-Leffler function (Prabhakar function)

The three-parameter Mittag-Leffler function (Prabhakar function) is defined by^{[1] [4] [5]}

E ^{\gamma} _{\alpha, \beta} (z) = \sum ^{inf ty} _{n =0} \dfrac{(\gamma) _{n}{n!\Gamma(\alpha} n + \beta)} z ^{n,} \Re(\alpha)> 0

where

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

Elementary special cases

The following special cases immediately follow from the definition.^[6]

E ^{0} _{\alpha,\beta} (z) = (1)/(\Gamma(\beta)) E ^{1} _{\alpha,\beta} (z) = E _{\alpha,\beta} (z) , the two-parameter Mittag-Leffler function.E ^{1} _{\alpha, 1} (z) = E _{\alpha} (z) , the one-parameter Mittag-Leffler function.E ^{1} _{1 , 1} (z) = e ^{z} , the classical exponential function.

Properties

Reduction formula

The following formula can be reduced to lower the value of the third parameter

\gamma .^[6] E ^{\gamma +1} _{\alpha,\beta} (z) = (1)/(\alpha\gamma) [ E ^{\gamma(z)} _{\alpha, \beta -1} + ( 1- \beta + \alpha\gamma)E ^{\gamma(z) ]} _{\alpha,\beta}

Relation with Fox–Wright function

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

E ^{\gamma(z)} _{\alpha, \beta} = (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

(d)/(dz) \left( E ^{\gamma(z)\right)} _{\alpha,\beta} = (1)/(\alpha z) [ E ^{\gamma(z)} _{\alpha, \beta -1} + ( 1- \beta)E ^{\gamma ]} _{\alpha, \beta}

There is a general expression for higher order derivatives. Let

m be a positive integer. The m -th derivative of the Prabhakar function is given by(d ^{m})/(dz ^{m}) \left( E ^{\gamma(z)\right)} _{\alpha,\beta} = (\Gamma(\gamma + m))/(\Gamma(\gamma)) E ^{\gamma + m} _{\alpha, m\alpha + \beta} (z)

The following result is useful in applications.

(d ^{m})/(dz ^{m}) \left( t ^{\beta-1} E ^{\gamma(t} _{\alpha,\beta} ^{\alpha} z) \right) = t ^{\beta} E ^{\gamma(t} _{\alpha, \beta - m} ^{\alpha} z)

Integrals

The following result involving Prabhakar function is known.

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

L \left[ t ^{\beta} E ^{\gamma} _{\alpha, \beta} (t ^{\alpha} z) ; s\right] = (s ^{\alpha\gamma-\beta})/((s ^{\alpha} - z) ^{\gamma}) , \Re(s)> 0 , |s|>|z| ^{1/\alpha}

Prabhakar fractional calculus

The following function is known as the Prabhakar kernel in the literature.^[6]

e ^{\gamma} _{\alpha,\beta} (t; λ ) = t ^{\beta} E ^{\gamma(t} _{\alpha,\beta} ^{\alpha} z)

Given any function

f(t) , the convolution of the Prabhakar kernel and f(t) is called the Prabhakar fractional integral:\int ^{t} _{t _{0}} (t - u) ^{\beta} E ^{\gamma} _{\alpha,\beta} \left( λ (t - u) ^{\alpha\right)} f(u) du

Properties of the Prabhakar fractional integral have been extensively studied in the literature.^{[7] [8]}

Notes and References

1. Tilak Raj Prabhakar . A singular integral equation with a generalized Mittag-Leffler function in the kernel . Yoklohama Mathematics Journal . 1971 . 19 . 1 . 7–15 . 27 December 2023.
2. Book: Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. Rogosin . Mittag-Leffler Functions, Related Topics and Applications . 2014 . Springer . 978-3-662-43929-6. 17.
3. Book: Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. Rogosin . Mittag-Leffler Functions, Related Topics and Applications . 2014 . Springer . 978-3-662-43929-6. 56.
4. Book: Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. Rogosin . Mittag-Leffler Functions, Related Topics and Applications . 2014 . Springer . 978-3-662-43929-6. 97.
5. Roberto Garra and Roberto Garrappa . The Prabhakar or three parameter Mittag–Leffler function: theory and application . Communications in Nonlinear Science and Numerical Simulation . 2018 . 56 . 314–329 . 10.1016/j.cnsns.2017.08.018 . 1708.07298v2 . 2018CNSNS..56..314G .
6. Andrea Giusti, Ivano Colombaro, Roberto Garra . A practical guide to Prabhakar fractional calculus . Fractional Calculus and Applied Analysis . 2020 . 25 . 1 . 9–54 . 10.1515/fca-2020-0002 . 2002.10978 .
7. Anatoly A. Kilbas, Megumi Saigo and R. K. Saxena . Generalized mittag-leffler function and generalized fractional calculus operators . Integral Transforms and Special Functions . 2004 . 15 . 1 . 31–49. 10.1080/10652460310001600717 . 120569191 .
8. F. Polito and Z. Tomovski . Some properties of Prabhakar-type fractional calculus operators . Fractional Differential Calculus . 2016 . 6 . 1 . 73–94. 10.7153/fdc-06-05 . 1508.03224 .