arXiv:1705.03762v1 [math.CA] 7 May 2017 An Extended k−type Hypergeometric Functions Praveen Agarwal Department of Mathematics, Anand International College of Engineering, Jaipur 303012, Rajasthan, India E-Mail: praveen.agarwal@anandice.ac.in goyal.praveen2011@gmail.com Mohamed Jleli Department of Mathematics, King Saud University, Riyadh-11451, Kingdom Saudi Arabia E-Mail: jleli@ksu.edu.sa Abstract Hypergeometric functions and their generalizations play an important rˆ oles in diverse applications. Many authors have been established gener- alizations of hypergeometric functions by a number ways. In this paper, we aim at establishing (presumably new) extended k−type hypergeomet- ric function 2 f k 1 [a, b; c; ω; z] and study various properties including integral representations, differential formulas and fractional integral and deriva- tive formula. 2010 Mathematics Subject Classification. Primary 33B15, 33C15, 33C20; Sec- ondary 33B20, 26A33. Key Words and Phrases. k−hypergeometric function; k− Gamma and Beta func- tions; Differential formulas; Integral representation; fractional integral and differential operator. 1. Introduction Many functions in sciences and engineering are important due to their own unique sent of properties which are popularly known as special functions(such as the Gamma and Beta functions, the Gauss hypergeometric function, and so on). In few decades many interesting and important extensions of the hypergeometric function have been established by several authors (see, for example, [2, 4, 5, 7, 9]). Our present study have largely motivated by the above-mentioned works. Recently, Diaz and Pariguan[4] introduced the k-gamma function Γ k (u)=  ∞ 0 e - x k k x u-1 dx = k k-1 Γ  u k  ℜ(u) > 0, (1.1) and the k-beta function B k (u, v)= 1 k  1 0 z u k -1 (1 − z) u k -1 dz, ℜ(u) > 0, ℜ(u) > 0, (1.2)