Paper: Analytic properties of some basic hypergeometric-Sobolev-type orthogonal polynomials

Analytic properties of some basic hypergeometric-Sobolev-type orthogonal polynomials

Abstract

In this contribution we consider sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product \(\langle f, g\rangle_S :=\langle {\bf u}, f\,g\rangle+N({\mathcal D}_q\, f)(\alpha)({\mathcal D}_q\, g)(\alpha)\), where \(\alpha\in \mathbb R, N\ge 0\), and \({\bf u}\) is a \(q\)-classical linear functional and \({\mathcal D}_q\) is the \(q\)-derivative operator. We obtain some algebraic properties of these polynomials such as an explicit representation, a five-term recurrence relation as well as a second order linear \(q\)-difference holonomic equation fulfilled by such polynomials.

We present an analysis of the behaviour of its zeros as a function of the mass \(N\). In particular, we obtain the exact values of \(N\) such that the smallest (respectively, the greatest) zero of the studied polynomials is located outside of the support of the measure. We conclude this work considering two examples.

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BibTeX

@article {MR3883341,
AUTHOR={Costas-Santos, R. S. and Soria-Lorente, A.},
TITLE={Analytic properties of some basic hypergeometric-{S}obolev-type orthogonal polynomials},
JOURNAL={J. Difference Equ. Appl.},
FJOURNAL={Journal of Difference Equations and Applications},
VOLUME={24},
YEAR={2018},
NUMBER={11},
PAGES={1715--1733},
ISSN={1023-6198},
MRCLASS={33D45 (05A30 39A13)},
MRNUMBER={3883341},
ZBL={3A1405.33024},
DOI={10.1080/10236198.2018.1517760},
URL={https://doi.org/10.1080/10236198.2018.1517760},
}