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},
}