Turán-Type Inequalities for Generalized k-Bessel Functions

Abstract

We propose an approach to the generalized k-Bessel function defined by

\({\text{U}}_{p,q,r}^{\text{k}}\left(z\right)=\sum_{n=0}^{\infty }\frac{{\left(-r\right)}^{n}}{{\Gamma }_{k}\left(nk+p+\frac{q+1}{2}\text{k}\right)n!}{\left(\frac{z}{2}\right)}^{2n+\frac{p}{\text{k}}},\)

where k > 0 and p, q, r ∈ \({\mathbb{C}}\). We discuss the uniform convergence of \({\text{U}}_{p,q,r}^{\text{k}}\) (z). Moreover, we prove that the analyzed function is entire and determine its growth order and type. We also find its Weierstrass factorization, which turns out to be an infinite product uniformly convergent on a compact subset of the complex plane. The integral representation for \({\text{U}}_{p,q,r}^{\text{k}}\) (z) is found by using the representation for k-beta functions. We also prove that the specified function is a solution of a second-order differential equation that generalizes certain well-known differential equations for the classical Bessel functions. In addition, some interesting properties, such as recurrence and differential relations, are demonstrated. Some of these properties can be used to establish Turán-type inequalities for this function. Ultimately, we study the monotonicity and log-convexity of the normalized form of the modified k-Bessel function \({\text{T}}_{p,q,1}^{\text{k}}\) defined by \({\text{T}}_{p,q,1}^{\text{k}}\) (z) = \(i{-}^\frac{p}{k}{\text{U}}_{p,q,1}^{\text{k}}\) (iz), as well as the quotient of the modified k-Bessel function, exponential, and k-hypergeometric functions. In this case, the leading concept of the proofs comes from the monotonicity of the ratio of two power series.