On the Zonal function and the Faraut-Koranyi hypergeometric function in rank two

In this paper we give a complete description of the generalized hypergeometric functions, introduced by Faraut and Kor\’{a}nyi on the Cartan domains. We establish some recursive relations with different arguments of zonal functions and some Gauss type contiguous relations between the Faraut-Kor\’{a}nyi hypergeometric functions on the domains of rank two. Finally, we give an infinite sums involving classical hypergeometric functions.