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 A new extension of Bessel function Commun. Korean Math. Soc.Published online December 16, 2020 MEERA H CHUDASAMA P. D. Patel Institute of Applied Sciences, Charotar University of Science & Technology Abstract : In this paper, we propose an extension of the classical Bessel function by means of our $\ell$-hypergeometric function [Annali Dell'Universit$\grave{a}$ Di Ferrara, \textbf{62}(1), 23--38 (2016)]. As the main results, the infinite order differential equation, the generating function relation, and Contour integral representations including Schl$\ddot{a}$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 Full-Text :