Paper: On polynomials orthogonal with respect to an inner product involving higher order difference: The Meixner case

On polynomials orthogonal with respect to an inner product involving higher order differences: The Meixner case

Abstract

In this contribution we consider sequences of monic polynomials orthogonal with respect to the Sobolev-type inner product \[ \left\langle f,g\right\rangle= \langle {\bf u}^{\tt M},fg\rangle +\lambda \mathscr T^j f (\alpha)\mathscr T^{j}g(\alpha), \] where \({\bf u}^{\tt M}\) is the Meixner linear operator, \(\lambda\in\mathbb{R}_{+}\), \(j\in\mathbb{N}\), \(\alpha \leq 0\), and \(\mathscr T\) is the forward difference operator \(\Delta\), or the backward difference operator \(\nabla\).

We derive an explicit representation for these polynomials. The ladder operators associated with these polynomials are obtained, and the linear difference equation of second order is also given. In addition, for these polynomials we derive a \((2j+3)\)-term recurrence relation. Finally, we find the Mehler-Heine type formula for the particular case \(\alpha=0\).

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BibTeX

@Article{cosovi,
AUTHOR={Costas-Santos, R. S. and Soria-Lorente, A. and Vilaire, J.-M.},
TITLE={On Polynomials Orthogonal with Respect to an Inner Product Involving Higher-Order Differences: The Meixner Case},
JOURNAL={Mathematics},
VOLUME={10},
YEAR={2022},
NUMBER={11},
ARTICLE-NUMBER={1952},
DOI={10.3390/math10111952},
URL={https://doi.org/10.3390/math10111952},
ISSN={2227-7390},
DOI={10.3390/math10111952},
}