On the variation with respect to a parameter of zeros of Bessel and q- Bessel functions
type: Item
instance of: scholarly article
title: On the variation with respect to a parameter of zeros of Bessel and q- Bessel functions
zbMATH Open document ID: 0651.33007
DOI: 10.1016/0022-247X(88)90148-5
author: Mourad E. H. Ismail
author: Martin E. Muldoon
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: We apply what is essentially the Hellmann-Feynman theorem in a discrete setting to derive representations for the derivatives with respect to a parameter of the positive zeros of a family of entire functions. These representations prove that the zeros are monotone functions of the parameters involved. This family includes the Bessel functions \(J_{\nu}(x)\) and their basic analog \(J_{\nu}^{(2)}(x;q)\). As a by- product we show that \(j^ 2_{\nu,1}/(\nu +1)\) increases with \(\nu\) when \(\nu\in (-1,\infty)\), \(j_{\nu,1}\) being the smallest positive zero of \(J_{\nu}(x)\). We also use the Lommel polynomials to derive improved lower bounds for the smallest positive zero of \(J_{\nu}(x)\).
Mathematics Subject Classification ID: 33C10
zbMATH DE Number: 4061652
zbMATH Keywords: Bessel functions
zbMATH Keywords: Lommel polynomials
MaRDI profile type: Publication
cites work: ON THE DEPENDENCE OF AN EIGENVALUE ON A PARAMETER
cites work: On a class of polynomials orthogonal over a denumerable set
cites work: Q5651013
cites work: Q3874513
cites work: On the concavity of zeros of bessel functions†
cites work: On the Square of the Zeros of Bessel Functions
cites work: On the Convexity of the Zeros of Bessel Functions
cites work: FURTHER RESULTS ON THE ZEROS OF BESSEL FUNCTIONS
cites work: Forces in Molecules
cites work: Elementary approximations for zeros of Bessel functions
cites work: Polynomials orthogonal over a denumerable set
cites work: A differential equation for the zeros of bessel functions
cites work: The zeros of basic Bessel functions, the functions J(nu+ax)(x), and associated orthogonal polynomials
cites work: On zeros of bessel functions†
cites work: The variation of zeros of certain orthogonal polynomials
cites work: Monotonicity of the Zeros of a Cross-Product of Bessel Functions
cites work: Q5526485
cites work: The Rayleigh Function
cites work: Inequalities and Approximations for Zeros of Bessel Functions of Small Order
cites work: Monotonicity and Convexity Properties of Zeros of Bessel Functions
cites work: On the Zeros of a Cross‐Product of Bessel Functions of Different Orders
cites work: Q3037780
cites work: A Differential Equations Proof of a Nicholson-Type Formula
cites work: Q3733181
cites work: A Note on the Rayleigh Polynomials
cites work: A Series Expansion for the First Positive Zero of the Bessel Functions
cites work: Q4769312
cites work: A class of continued fractions
cites work: Q3849152
cites work: Some theorems on perturbation theory
cites work: Q5846799
full work available at URL: 0022-247x(88)90148-5
OpenAlex ID: W2042587139
description: scientific article; zbMATH DE number 4061652