An Extended k-type Hypergeometric Functions

Hypergeometric functions and their generalizations play an important r^{o}les in diverse applications. Many authors have been established generalizations of hypergeometric functions by a number ways. In this paper, we aim at establishing (presumably new) extended $k-$type hypergeometric function ${}_2{f}_1^k[a,b;c;\omega ;z]$ and study various properties including integral representations, differential formulas and fractional integral and derivative formula.