Volume 38, pp. 317-326, 2011.

Stieltjes interlacing of zeros of Jacobi polynomials from different sequences

K. Driver, A. Jooste, and K. Jordaan

Abstract

A theorem of Stieltjes proves that, given any sequence $\{p_n{\}}_{n=0}^{\mathrm{\infty }}$ of orthogonal polynomials, there is at least one zero of $p_n$ between any two consecutive zeros of $p_k$ if $k<n$, a property called Stieltjes interlacing. We show that Stieltjes interlacing extends, under certain conditions, to the zeros of Jacobi polynomials from different sequences. In particular, we prove that the zeros of $P_{n+1}^{\alpha ,\beta }$ interlace with the zeros of $P_{n-1}^{\alpha +k,\beta }$ and with the zeros of $P_{n-1}^{\alpha ,\beta +k}$ for $k\in \{1,2,3,4\}$ as well as with the zeros of $P_{n-1}^{\alpha +t,\beta +k}$ for $t,k\in \{1,2\}$; and, in each case, we identify a point that completes the interlacing process. More generally, we prove that the zeros of the $k$th derivative of $P_n^{\alpha ,\beta }$, together with the zeros of an associated polynomial of degree $k$, interlace with the zeros of $P_{n+1}^{\alpha ,\beta },k,n\in N,k<n$.

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Key words

Interlacing of zeros; Stieltjes’ Theorem; Jacobi polynomials.

AMS subject classifications

33C45, 42C05

ETNA articles which cite this article

Vol. 44 (2015), pp. 271-280 Kenier Castillo and Fernando R. Rafaeli: On the discrete extension of Markov's theorem on monotonicity of zeros