## 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 $\{p_n\}_{n=0}^\infty$ 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 $P_{n+1}^{\alpha,\beta}, k,n \in N, k < n$.

Full Text (PDF) [133 KB], BibTeX

### Key words

Interlacing of zeros; Stieltjes’ Theorem; Jacobi polynomials.

33C45, 42C05