Tur\'an determinants of Bessel functions

TL;DR

This paper surveys and extends Turán inequalities for Bessel functions, introduces new integral representations, and proves a new Turán inequality for Bessel functions of the second kind, advancing understanding of their properties.

Contribution

It extends higher order Turán inequalities to real parameters and derives new integral formulas, also establishing a Turán inequality for Bessel functions of the second kind.

Findings

Extended Turán inequalities to real parameters.

Derived new integral representation formulas.

Proved a Turán inequality for Bessel functions of the second kind.

Abstract

In this paper first we survey the Tur\'an type inequalities and related problems for the Bessel functions of the first kind. Then we extend the known higher order Tur\'an type inequalities for Bessel functions of the first kind to real parameters and we deduce new closed integral representation formulae for the second kind Neumann type series of Bessel functions of the first kind occurring in the study of Tur\'an determinants of Bessel functions of the first kind. At the end of the paper we prove a Tur\'an type inequality for the Bessel functions of the second kind.