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Published in:
04-01-2020 | Original Paper
Orthogonal sequences constructed from quasi-orthogonal ultraspherical polynomials
Published in: Numerical Algorithms | Issue 2/2020
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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 < λ < − 1/2, the two curves have a similar shape while the curves are almost identical for λ > − 1/2.