arXiv subjects

K. Jordaan

Publications and source records attributed to K. Jordaan.

4 recordsLinked to original sources

arXiv2024

Bounds for extreme zeros of Meixner-Pollaczek polynomials

In this paper we consider connection formulae for orthogonal polynomials in the context of Christoffel transformations for the case where a weight function, not necessarily even, is multiplied by an even function $c_{2k}(x),k\in N_0$, to determine new lower bounds for the largest zero and upper bounds for the smallest zero of a Meixner-Pollaczek polynomial. When $p_n$ is orthogonal with respect to a weight $w(x)$ and $g_{n-m}$ is orthogonal with respect to the weight $c_{2k}(x)w(x)$, we show that $k\in\{0,1,\dots,m\}$ is a necessary and sufficient condition for existence of a connection formula involving a polynomial $G_{m-1}$ of degree $(m-1)$, such that the $(n-1)$ zeros of $G_{m-1}g_{n-m}$ and the $n$ zeros of $p_n$ interlace. We analyse the new inner bounds for the extreme zeros of Meixner-Pollaczek polynomials to determine which bounds are the sharpest. We also briefly discuss bounds for the zeros of Pseudo-Jacobi polynomials.

math.CA↗

arXiv2023

On zeros of quasi-orthogonal Meixner polynomials

For each fixed value of $β$ in the range $-2<β<-1$ and $0<c<1$, we investigate interlacing properties of the zeros of polynomials of consecutive degree for $M_{n}(x;β,c)$ and $M_k(x,β+t,c)$, $k\in\{n-1,n,n+1\}$ and $t\in\{0,1,2\}$. We prove the conjecture in [K. Driver and A. Jooste, Quasi-orthogonal Meixner polynomials, Quaest. Math. 40 (4) (2017), 477-490] on a lower bound for the first positive zero of the quasi-orthogonal order $1$ polynomial $M_n(x;β+1,c)$ and identify upper and lower bounds for the first few zeros of quasi-orthogonal order $2$ Meixner polynomials $M_n(x;β,c)$. We show that a sequence of Meixner polynomials $\{M_n(x;β,c)\}_{n=3}^{\infty}$ with $-2<β<-1$ and $0<c<1$ cannot be orthogonal with respect to any positive measure by proving that the zeros of $M_{n-1}(x;β,c)$ and $M_{n}(x;β,c)$ do not interlace for any $n\in\mathbb{N}_{\geqq 3}.$

math.CA↗

arXiv2013

Real zeros of 2F1 hypergeometric polynomials

We use a method based on the division algorithm to determine all the values of the real parameters $b$ and $c$ for which the hypergeometric polynomials $_2F_1(-n, b; c; z)$ have $n$ real, simple zeros. Furthermore, we use the quasi-orthogonality of Jacobi polynomials to determine the intervals on the real line where the zeros are located.

math.CA↗

arXiv2011

Bounds for extreme zeros of some classical orthogonal polynomials

We derive upper bounds for the smallest zero and lower bounds for the largest zero of Laguerre, Jacobi and Gegenbauer polynomials. Our approach uses mixed three term recurrence relations satisfied by polynomials corresponding to different parameter(s) within the same classical family. We prove that interlacing properties of the zeros impose restrictions on the possible location of common zeros of the polynomials involved and deduce strict bounds for the extreme zeros of polynomials belonging to each of these three classical families. We show numerically that the bounds generated by our method improve known lower (upper) bounds for the largest (smallest) zeros of polynomials in these families, notably in the case of Jacobi and Gegenbauer polynomials.

math.CA↗