Properties of the Tur\'anian of modified Bessel functions

TL;DR

This paper introduces new series and integral representations for the Turanian of modified Bessel functions, providing improved bounds, asymptotic expansions, and compact forms that enhance understanding of their properties.

Contribution

It presents novel representations and bounds for the Turanian of modified Bessel functions, improving upon existing results and offering a more uniform understanding for natural and real orders.

Findings

New series and integral representations derived.

Established tighter bounds and asymptotic expansions.

Provided a compact form leading to a uniform upper bound.

Abstract

In this paper some new series and integral representations for the Tur\'anian of modified Bessel functions of the first kind are given, which give new asymptotic expansions and tight bounds for the Tur\'an determinant in the question. It is shown that in the case of natural and real order the Tur\'anian can be represented in a relatively compact form, which yields a uniform upper bound for the Tur\'an determinant for modified Bessel functions of the first kind. Our results complement and improve some of the results from the literature.