The New Mittag-Leffler Function and Its Applications

Abstract

In this paper, we investigate some properties of the Pochhammer $\left(p,s,k\right)$ -symbol ${}_p{{\displaystyle \left[\xi \right]}}_{n,k,s}$ and gamma $\left(p,s,k\right)$ -function ${}_p{{\displaystyle \mathrm{\Gamma }}}_{s,k}\left(\xi \right)$ . We then prove several identities for newly defined symbol ${}_p{{\displaystyle \left[\xi \right]}}_{n,k,s}$ and the function ${}_p{{\displaystyle \mathrm{\Gamma }}}_{s,k}\left(\xi \right)$ . The integral representations for the gamma $\left(p,s,k\right)$ -function and beta $\left(p,s,k\right)$ -function are presented. Also, we define a new Mittag-Leffler $\left(p,s,k\right)$ -function and study its analytic properties and its transforms.