Communications of the
Korean Mathematical Society CKMS

In this paper, we obtain a $(p,q)$-extension of the Whittaker function $M_{k,\mu}(z)$ together with its integral representations, by using the extended confluent hypergeometric function of the first kind $\Phi_{p,q}(b;c;z)$ [recently extended by J. Choi]. Also, we give some of its main properties, namely the summation formula, a transformation formula, a Mellin transform, a differential formula and inequalities. In addition, our extension on Whittaker function finds interesting connection with the Laguerre polynomials.

Keywords: Whittaker function, extended Beta function, Mellin transform, extended confluent hypergeometric function, Laguerre polynomials, extended Gauss hypergeometric function

MSC numbers: 33C15, 33B15, 33C05, 35A22, 33C45