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