Interlacing of the zeros of contiguous hypergeometric functions
type: Item
instance of: scholarly article
title: Interlacing of the zeros of contiguous hypergeometric functions
zbMATH Open document ID: 1171.33008
author: Javier Segura
published in: Numerical Algorithms
publication date: 2009-04-29T00:00:00Z
review text: Interlacing properties of the zeros of hypergeometric and confluent hypergeometric functions (Bessel functions and Hermite, Laguerre and Jacobi polynomials) are explored. The main result to be applied for proving interlacing is the following statement (Lemma 1), proved in [\textit{J. Segura}, SIAM J. Numer. Anal. 40, No. 1, 114--133 (2002; Zbl 1058.33020)] in a slightly different version: Let \(y_m(x)\) and \(y_{m-1} (x)\) be two nontrivial solutions of the system of first order difference-differential equations (DDEs) \[ y'_{m}(x)=a_m(x)y_{m}(x)+d_m(x)y_{m-1}(x),\qquad y'_{m-1}(x)= b_m(x)y_{m-1}(x)+e_m(x)y_{m}(x), \] with continuous coefficients, as functions on \(x\) in an interval \(I\) and such that \(d_m(x)\) and \(e_m(x)\) do not change the sign in \(I\). If one of these functions \(y_m(x)\) or \(y_{m-1} (x)\) has, at least, two zeros in \(I\), then the zeros of \(y_m(x)\) and \(y_{m-1} (x)\) are interlaced and \(d_m(x)e_m(x)<0\) in \(I\). A sequence of hypergeometric functions \(y_m(x)={}_2F_1 (a+ \varepsilon_1m, b+\varepsilon_2m;c+\varepsilon_3m;x)\) (\(\varepsilon_i, m\in\mathbb Z\), \(a,b,c,x\in\mathbb R\)) satisfy first order DDEs with coefficients continuous in \((0,1)\). Using the above statement, in particular, it is proved that if \(p_{n+1}(x)\) and \(p_{n- 1}(x)\) are two classical orthogonal polynomials (Hermite, Laguerre, Jacobi) with respect to the same weight function \(w(x)\) in \([a,b]\), then the zeros of \(p_{n+1}(x)\) and \(p_{n- 1}(x)\) are interlaced for \(x>\beta_n\) and \(x<\beta_n\), where \(\beta_n=\int_a^bxp^2_n(x)w(x)\,dx/\int_a^bp^2_n(x) w(x)\,dx\in(a,b)\) (Theorem 1).
reviewed by: Valery V. Karachik
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33C05
Mathematics Subject Classification ID: 34K06
zbMATH DE Number: 5549495
zbMATH Keywords: hypergeometric functions
zbMATH Keywords: first order DDEs
zbMATH Keywords: zeros
zbMATH Keywords: interlacing
describes a project that uses: SumTools
MaRDI profile type: Publication
full work available at URL: s11075-008-9211-x
OpenAlex ID: W2056477932
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DOI: 10.1007/S11075-008-9211-X
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description: scientific article; zbMATH DE number 5549495