Honam Mathematical J. 38 (2016), No. 2, pp. 325–335 http://dx.doi.org/10.5831/HMJ.2016.38.2.325 A NOTE ON GENERALIZED EXTENDED WHITTAKER FUNCTION Nabiullah Khan and Mohd Ghayasuddin * Abstract. In the present paper, we define the generalized extended Whittaker function in terms of generalized extended confluent hy- pergeometric function of the first kind. We also study its integral representation, some integral transforms and its derivative. 1. Introduction Many extensions of special functions have been given by a number of authors, namely, Chaudhry et al. [9], Chaudhry et al. [10], Lee et al. [2], Parmar [12], Liu and Wang [8] and Khan and Ghayasuddin [11]. Motivated by the above-mentioned works, in the present paper, we present a new generalization of Whittaker function of first kind in terms of the generalized extended confluent hypergeometric function. For our present study, we recall here the following extensions of special functions. The classical beta function, denoted by B(a, b) and is defined (see [13], see also [4]) by the Euler’s integral B(a, b)=  1 0 t a-1 (1 − t) b-1 dt, = Γ(a) Γ(b) Γ(a + b) (ℜ(a) > 0, ℜ(b) > 0) . (1.1) Received August 28, 2015. Accepted March 7, 2016. 2010 Mathematics Subject Classification. 33B15, 33C05, 33C15. Key words and phrases. Beta function, Extended beta function, Confluent hy- pergeometric function, Extended confluent hypergeometric function, Gauss hyperge- ometric function, Extended Gauss hypergeometric function. *Corresponding author.