Orthogonal sequences constructed from quasi-orthogonal ultraspherical polynomials

Abstract

Let $\{x_{k,n-1}{\}}_{k=1}^{n-1}$ and $\{x_{k,n}{\}}_{k=1}^n,$ $n\in \mathbb{N}$, be two sets of real, distinct points satisfying the interlacing property $x_{i,n}<x_{i,n-1}<x_{i+1,n},i=1,2,\dots ,n-1.$ Wendroff proved that if $p_{n-1}(x)=\prod _{k=1}^{n-1}(x-x_{k,n-1})$ and $p_n(x)=\prod _{k=1}^n(x-x_{k,n})$, then $p_{n-1}$ and $p_n$ can be embedded in a non-unique monic orthogonal sequence $\{p_n{\}}_{n=0}^{\infty }.$ We investigate a question raised by Mourad Ismail at OPSFA 2015 as to the nature and properties of orthogonal sequences generated by applying Wendroff's Theorem to the interlacing zeros of $C_{n-1}^{\lambda }(x)$ and $(x^2-1)C_{n-2}^{\lambda }(x)$, where $\{C_k^{\lambda }(x){\}}_{k=0}^{\infty }$ is a sequence of monic ultraspherical polynomials and $-3/2<\lambda <-1/2,$ $\lambda \ne -1.$ We construct an algorithm for generating infinite monic orthogonal sequences $\{D_k^{\lambda }(x){\}}_{k=0}^{\infty }$ from the two polynomials $D_n^{\lambda }(x):=(x^2-1)C_{n-2}^{\lambda }(x)$ and $D_{n-1}^{\lambda }(x):=C_{n-1}^{\lambda }(x)$, which is applicable for each pair of fixed parameters $n,\lambda$ in the ranges $n\in \mathbb{N},n\ge 5$ and $\lambda >-3/2$, $\lambda \ne -1,0,(2k-1)/2,k=0,1,\dots$. We plot and compare the zeros of $D_m^{\lambda }(x)$ and $C_m^{\lambda }(x)$ for several choices of $m\in \mathbb{N}$ and a range of values of the parameters $\lambda$ and $n$. For $-3/2<\lambda <-1,$ the curves that the zeros of $D_m^{\lambda }(x)$ and $C_m^{\lambda }(x)$ approach are substantially different for large values of $m.$ When $-1<\lambda <-1/2,$ the two curves have a similar shape while the curves are almost identical for $\lambda >-1/2.$