Siago’s $K$-Fractional Calculus Operators

Anjali Gupta a and C.L. Parihar b, *

Siago’s $K$-Fractional Calculus Operators

Authors :

Anjali Gupta ^a and C.L. Parihar^{b, *}

Author Address :

^a1523, Sudama Nagar, 60 Feet road, Indore-452009, M.P, India.
^b 500-Pushpratan Park, devguradia, Indore-452016, M.P, India.

*Corresponding author.

Abstract :

The aim of present paper is to define a pair of $k$--Saigo fractional integral and derivative operators involving generalized $k$--hypergeometric function. The Saigo-$k$ generalized fractional operators involving $k$--hypergeometric function in the kernel are applied to the generalized $k$--Mittag-Leffler function and evaluate the formula
egin{align*}
_2 F_{1, k}
left [
egin{tabular}{cc}
($al, k), (e, k)$ & \
& ; $frac{1}{k}$ \
($ga, k)$ &
end{tabular}
ight ] = dis frac{Ga_k(ga) Ga_k{(ga-al-e)}}{ Ga_k(ga-al)Ga_k(ga-e)}
end{align*}
using the integral representation for $k$-hypergeometric function.

Keywords :

$k$-functions and $k$-fractional calculus.

DOI :

Article Info :

Received : January 07, 2017; Accepted : May 23, 2017.