Bounds for the small real and purely imaginary zeros of Bessel and related functions
type: Item
instance of: scholarly article
title: Bounds for the small real and purely imaginary zeros of Bessel and related functions
zbMATH Open document ID: 0845.33003
author: Mourad E. H. Ismail
author: Martin E. Muldoon
published in: Methods and Applications of Analysis
publication date: 1995-09-20T00:00:00Z
review text: Denote by \(J_\nu (z)\) the Bessel function of order \(\nu\). Since \(z^{- \nu} J_\nu (z)\) is an even entire function its zeros are located symmetrically with respect to both the real and the imaginary axis. Denote by \(z_{\nu 1}\) the smallest positive zero of \(J_\nu (z)\) for \(\nu > - 1\). As \(\nu \to - 1 + \), \(z_{\nu 1}\) and \(- z_{\nu 1}\) collapse at the origin and for \(-1 > \nu < - 2\), they split again into two imaginary zeros \(z_{\nu 1}\) and \(- z_{\nu 1}\). The aim of this paper is to give bounds for \(z^2_{\nu 1} (< 0)\), \(- 2 < \nu < - 1\) and \(z^2_{\nu 1} (> 0)\), \(\nu > - 1\) in terms of \(\nu\). Two methods are used for this purpose. The first one is going back to Euler and Rayleigh and uses the power sums of the zeros and Rayleigh's inequalities, the second one is based on continued fractions of \({zJ_{\nu - 1} (z) \over J_\nu (z)}\) and Lommel polynomials \(R_{n, \nu}\). In order to give a flavour of the results proved, consider the following inequalities shown among others in the paper: \[ \begin{aligned} 2 (\nu + 1) (\nu + 3) & < z^2_{\nu 1} < 4 (\nu + 1) (\nu + 2) \text{ for } - 2 < \nu < - 1 \quad \text{and} \\ z^2_{\nu 1} & < x^2_{n, \nu + 1}, \text{ for } n \geq 2,\;\nu > - 1 \quad \text{and} \\ - y^2_{2n - 1, \nu + 1} & < z^2_{\nu 1} < - y^2_{2n, \nu + 1}, \text{ for } n \geq 2,\;- 2 < \nu < - 1, \end{aligned} \] where \(x_{n, \nu + 1}\), \(y_{n, \nu + 1}\) denote the smallest positive zero of \(R_{n, \nu + 1} (x)\) and \(R_{n, \nu + 1} (iy)\) respectively. Related functions like \(\alpha J_\nu (z) + zJ_\nu' (z)\), \(J_\nu' (z)\), \(J_{\nu}'' (z)\) are investigated using the same approaches.
reviewed by: Ulrich Stadtmüller
Mathematics Subject Classification ID: 33C10
zbMATH DE Number: 798941
zbMATH Keywords: inequalities
zbMATH Keywords: Bessel function
zbMATH Keywords: zeros
zbMATH Keywords: Lommel polynomials
MaRDI profile type: Publication
full work available at URL: maa.1995.v2.n1.a1
OpenAlex ID: W179017527
DOI: 10.4310/MAA.1995.V2.N1.A1
description: scientific article; zbMATH DE number 798941