On the zeros of Jacobi polynomials
For \(\alpha>-1\), \(\beta>-1\), let \(x_{ni}= x_{ni} (\alpha, \beta)\) denote the \(i\)-th zero, in decreasing order, of the Jacobi polynomial \(P_ n^{(\alpha, \beta)} (x)\): \[ 1> x_{n1}> x_{n2}> \dots >x_{nn} >-1. \] Here the authors present a procedure based on the Sturm comparison theorem to obtain the following main result for \(x_{ni}\). For \(i=1,2, \dots,n\), let \(x_{ni}= x_{ni} (\alpha, \beta)\) be the \(i\)-th zero in decreasing order, of the Jacobi polynomial \(P_ n^{(\alpha, \beta)} (x)\). Then for \(\alpha >-{1\over 2}\), \(\beta> - {1\over 2}\) the following inequalities \[ \xi_ i< x_{ni}< \xi_{i-1}, \qquad i=1,2, \dots,n \] hold, where \(\xi_ 0, \xi_ 1,\dots, \xi_ n\) are the zeros of the function \[ (1- x^ 2)^{1/2} [(a-x) (x-b)]^{- 1/4} \sin \varphi(x) \] such that \[ \varphi(x)= (2n+ \overline {\alpha}+ \overline {\beta}) \arctan t- \overline {\alpha} \arctan \biggl( \sqrt {{{1-b} \over {1-a}} t} \biggr)- \overline {\beta} \arctan \biggl( \sqrt {{{1+b} \over {1+a}} t} \biggr) \] and \(\overline {\alpha}= \alpha+ {1\over 2}\) and \(\overline {\beta}= \beta+ {1\over 2}\).
scientific article; zbMATH DE number 638210