Interlacing of zeros of Gegenbauer polynomials of non-consecutive degree from different sequences

Abstract

A theorem due to Stieltjes’ states that if ${\{p_n\}_{n=0}^\infty}$ is any orthogonal sequence then, between any two consecutive zeros of p _k , there is at least one zero of p _n whenever k < n, a property called Stieltjes interlacing. We show that Stieltjes interlacing extends to the zeros of Gegenbauer polynomials ${C_{n+1}^{\lambda}}$ and ${C_{n-1}^{\lambda+t}}$, ${\lambda > -\frac 12}$, if 0 < t ≤ k + 1, and also to the zeros of ${C_{n+1}^{\lambda}}$ and ${C_{n-2}^{\lambda +k}}$ if ${k\in\{1,2,3\}}$. More generally, we prove that Stieltjes interlacing holds between the zeros of the kth derivative of ${C_{n}^{\lambda}}$ and the zeros of ${C_{n+1}^{\lambda}}$, ${k\in\{1,2,\dots,n-1\}}$ and we derive associated polynomials that play an analogous role to the de Boor–Saff polynomials in completing the interlacing process of the zeros.