Orthogonal sequences constructed from quasi-orthogonal ultraspherical polynomials

Abstract

Let $\displaystyle \{x_{k,n-1}\}_{k=1}^{n-1}$ and $\displaystyle \{x_{k,n}\}_{k=1}^{n},$ $n \in \mathbb {N}$ , 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$ . In [15], Wendroff proved that if $p_{n-1}(x) = \displaystyle \prod \limits _{k=1}^{n-1} (x-x_{k,n-1})$ and $p_{n}(x) = \displaystyle \prod \limits _{k=1}^{n} (x-x_{k,n})$ , then pn− 1 and pn can be embedded in a non-unique monic orthogonal sequence $\{p_{n}\}_{n=0}^{\infty }. $ We investigate a question raised by Mourad Ismail 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 < λ < − 1/2, λ≠ − 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, λ in the ranges $n \in \mathbb {N}, n \geq 5$ and λ > − 3/2, λ≠ − 1,0,(2k − 1)/2, k = 0,1,…. We plot and compare the zeros of $D_{m}^{\lambda } (x)$ and $C_{m}^{\lambda } (x)$ for selected choices of $m \in \mathbb {N}$ and a range of values of the parameters λ and n. For − 3/2 < λ < − 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. In contrast, when − 1 − 1/2.