A Study of Appell Function of Two Variables Using Hurwitz - Lerch Zeta Function of Two Variables

Abstract

Firstly, the Appell's Hypergeometric Function of two variables $F_{2}[a, b_{1}, b_{2};c_{1},c_{2};x_{1},x_{2}]$ is introduced using Hurwitz-Lerch Zeta Function of two variables $\phi_{a_1,a_2,b_1,b_2;c_1c_2} (x_1, x_2, s, p)$. Then several integral representations and differential formula are investigated for this function $F_{2}[a, b_{1},b_{2};c_{1},c_{2};x_{1},x_{2}]$. The function $F_{2}[a,b_{1},b_{2};c_{1},c_{2};x_{1},x_{2}]$ that we have introduced \& defined here has also been represented in term of generalized Hypergeometric Function ${}_{p}F_{q}$. To strengthen our main results, we have also considered some important special cases.