Zeros of Quasi-Orthogonal Jacobi Polynomials

Abstract

We consider interlacing properties satisfied by the zeros of Jacobi polynomials in quasi-orthogonal sequences characterised by $\alpha >-1$, $-2<\beta <-1$. We give necessary and sufficient conditions under which a conjecture by Askey, that the zeros of Jacobi polynomials $P_n^{(\alpha ,\beta )}$ and $P_n^{(\alpha ,\beta +2)}$ are interlacing, holds when the parameters $\alpha$ and $\beta$ are in the range $\alpha >-1$ and $-2<\beta <-1$. We prove that the zeros of $P_n^{(\alpha ,\beta )}$ and $P_{n+1}^{(\alpha ,\beta )}$ do not interlace for any $n\in \mathbb{N}$, $n\ge 2$ and any fixed $\alpha$, $\beta$ with $\alpha >-1$, $-2<\beta <-1$. The interlacing of zeros of $P_n^{(\alpha ,\beta )}$ and $P_m^{(\alpha ,\beta +t)}$ for $m,n\in \mathbb{N}$ is discussed for $\alpha$ and $\beta$ in this range, $t\ge 1$, and new upper and lower bounds are derived for the zero of $P_n^{(\alpha ,\beta )}$ that is less than $-1$.

References in corpus (1)

• Zeros of orthogonal polynomials generated by the Geronimus perturbation of measures

Cited by in corpus (5)

• On certain properties of perturbed Freud-type weight: a revisit
• Quasi-Orthogonality of Some Hypergeometric and $q$-Hypergeometric Polynomials
• Ratios of the Gauss hypergeometric functions with parameters shifted by integers: part I
• Zeros of quasi-orthogonal $q$-Laguerre polynomials
• Solvable dynamical systems and isospectral matrices defined in terms of the zeros of orthogonal or otherwise special polynomials