Zeros of polynomials orthogonal with respect to a signed weight

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 [1]; we prove that the interlacing property does not hold properly for such polynomial sequence.

[1] Atia M. J., Marcellan F., and Rocha I. A., On semi-classical orthogonal polynomials: A quasi-definite functional of class 1. Facta Universitatis (Nis). Ser. Math. Inform. 17 (2002), 25 —46