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 $W_{\nu ,c}^k\left(x\right):=\sum _{r=0}^{\infty }\frac{{(-c)}^r}{{\Gamma }_k(rk+\nu +k)r!}{\left(\frac{x}{2}\right)}^{2r+\frac{\nu }{k}}.$ We also indicate some representation formulae for the function introduced. Further, we show that the function $W_{\nu ,c}^k$ is a solution of a second-order differential equation. We investigate monotonicity and log-convexity properties of the generalized k-Bessel function $W_{\nu ,c}^k$ , particularly, in the case $c=-1$ . We establish several inequalities, including a Turán-type inequality. We propose an open problem regarding the pattern of the zeroes of $W_{\nu ,c}^k$ .