Zeros of orthogonal polynomials generated by the Geronimus perturbation of measures
Abstract
This paper deals with monic orthogonal polynomial sequences (MOPS in short) generated by a Geronimus canonical spectral transformation of a positive Borel measure $\mu$, i.e., \begin{equation*} \frac{1}{(xc)}d\mu (x)+N\delta (xc), \end{equation*} for some free parameter $N \in \mathbb{R}_{+}$ and shift $c$. We analyze the behavior of the corresponding MOPS. In particular, we obtain such a behavior when the mass $N$ tends to infinity as well as we characterize the precise values of $N$ such the smallest (respectively, the largest) zero of these MOPS is located outside the support of the original measure $\mu$. When $\mu$ is semiclassical, we obtain the ladder operators and the second order linear differential equation satisfied by the Geronimus perturbed MOPS, and we also give an electrostatic interpretation of the zero distribution in terms of a logarithmic potential interaction under the action of an external field. We analyze such an equilibrium problem when the mass point of the perturbation $c$ is located outside of the support of $\mu$.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.6256
 Bibcode:
 2014arXiv1402.6256B
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 30C15
 EPrint:
 Some minor typos in the text have been corrected. B. Murgante et al. (Eds.): ICCSA 2014  Lecture Notes in Computer Science (LNCS), 8579 (Part I) (2014), 4459. Springer International Publishing Switzerland. ISSN 03029743. ISBN13: 9783319091433. ISBN10: 3319091433