Zeros of polynomials orthogonal with respect to a signed weight

TL;DR

This paper studies the zeros of a specific polynomial sequence orthogonal with respect to a signed weight, providing new recurrence coefficients, analyzing zero interlacing properties, and establishing inequalities among the largest zeros.

Contribution

It introduces a novel method for deriving recurrence coefficients and analyzes zero distribution properties for polynomials orthogonal with a signed weight.

Findings

Recurrence coefficients are explicitly obtained using a new method.

Interlacing property of zeros does not hold properly for these polynomials.

Largest zeros satisfy specific inequalities depending on parameters.

Abstract

In this paper we consider the polynomial sequence $(P_{n}^{\alpha,q}(x))$ that is orthogonal on $[-1,1]$ with respect to the weight function $x^{2q+1}(1-x^{2})^{\alpha}(1-x), \alpha>-1, q\in \mathbb N$; we obtain the coefficients of the tree-term recurrence relation (TTRR) by using a different method from the one derived in \cite{kn:atia1}; we prove that the interlacing property does not hold properly for $(P_n^{\alpha,q}(x))$; and we also prove that, if $x_{n,n}^{\alpha+i,q+j}$ is the largest zero of $P_{n}^{\alpha+i,q+j}(x)$, $\displaystyle x_{2n-2j,2n-2j}^{\alpha+j,q+j}< x_{2n-2i,2n-2i}^{\alpha+i,q+i}, 0\leq i<j\leq n-1$.