Representation Formulae and Monotonicity of the Generalized k-Bessel Functions

Abstract

This paper introduces and studies a generalization of the $\mathtt{k}$-Bessel function of order $\nu$ given by \[\mathtt{W}^{\mathtt{k}}_{ν, c}(x):= \sum_{r=0}^\infty \frac{(-c)^r}{Γ_{\mathtt{k}}\left( r \mathtt{k} +ν+\mathtt{k}\right) r!} \left(\frac{x}{2}\right)^{2r+\fracν{\mathtt{k}}}. \] Representation formulae are derived for ${\mathtt{W}}_{\nu ,c}^{\mathtt{k}}.$ Further the function ${\mathtt{W}}_{\nu ,c}^{\mathtt{k}}$ is shown to be a solution of a second order differential equation. Monotonicity and log-convexity properties for the generalized $\mathtt{k}$-Bessel function ${\mathtt{W}}_{\nu ,c}^{\mathtt{k}}$ are investigated, particularly in the case $c=-1$. Several inequalities, including the Turán-type inequality are established.

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Cited by in corpus (2)

• Certain unified integration formulas associated with generalized k-Bessel function
• Generalized fractional integration of k\-Bessel function