Abstract
In this paper, we introduce and study a generalization of the k-Bessel function of order v given by
W-v,c(k)(x) := Sigma(infinity)(r=0) (-c)(r)/Gamma(k)(rk + v + k)r!(x/2)(2r+v/k).
We also indicate some representation formulae for the function introduced. Further, we show that the function W-v,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-v,c(k), particularly, in the case c = -1. We establish several inequalities, including a Turan-type inequality. We propose an open problem regarding the pattern of the zeroes of W-v,c(k).