Communications of the
Korean Mathematical Society CKMS

In this paper, we propose an extension of the classical Bessel function by means of our $\ell$-hypergeometric function \cite{mhc-bid2015}. As the main results, the infinite order differential equation, the generating function relation, and contour integral representations including Schl\"{a}fli's integral analogue are derived. With the aid of these, other results including some inequalities are also obtained. At the end, the graphs of these functions are plotted using the Maple software.

Keywords: Hypergeometric function, Bessel function, generating function relation, contour integral, infinite order operator

MSC numbers: 33C20, 33B10, 34A35