Paper: Zeros of polynomials orthogonal with respect to a signed weight

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

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BibTeX

@article {MR2877399,
AUTHOR={Atia, M. J. and Benabdallah, M. and Costas-Santos, R. S.},
TITLE={Zeros of polynomials orthogonal with respect to a signed weight},
JOURNAL={Indag. Math. (N.S.)},
FJOURNAL={Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae. New Series},
VOLUME={23},
YEAR={2012},
NUMBER={1-2},
PAGES={26--31},
ISSN={0019-3577},
MRCLASS={33C45 (42C05)},
MRNUMBER={2877399 (2012m:33007)},
ZBL={1236.42020},
MRREVIEWER={Serhan Varma},
DOI={10.1016/j.indag.2011.09.011},
URL={http://dx.doi.org/10.1016/j.indag.2011.09.011},
}