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.

We study a family of harmonic univalent functions using an operator involving q{derivative and Hypergeometric functions. We then obtain necessary and su, cient condition bounds for functions in this family. Extreme points and convex set for such functions are also introduced.

Suppose an urn contains M balls, of different types, which are removed from the urn in a uniform random manner. In the Hypergeometric coupon collection problem, we are interested in the set of balls that have been removed at the moment when at least one ball of each type has been removed. In its dual, we are interested in the set of removed balls at the first moment that this set contains all of the balls of at least one type.