Differential equation and inequalities of the generalized k-Bessel functions

In this paper, we introduce and study a generalization of the k-Bessel function of order ν given by [Formula: see text] We also indicate some representation formulae for the function introduced. Further, we show that the function [Formula: see text] is a solution of a second-order differential equation. We investigate monotonicity and log-convexity properties of the generalized k-Bessel function [Formula: see text] , particularly, in the case [Formula: see text] . We establish several inequalities, including a Turán-type inequality. We propose an open problem regarding the pattern of the zeroes of [Formula: see text] .