Prabhakar function

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 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.

[2] The one-parameter and two-parameter Mittag-Leffler functions are defined first.

is the well known gamma function defined by In the following it will be assumed that

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

-th derivative of the Prabhakar function is given by The following result is useful in applications.

The following result involving Prabhakar function is known.

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

The following function is known as the Prabhakar kernel in the literature.