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Talk: Sequentially ordered balanced Laguerre-Sobolev type orthogonal polynomials

Sequentially ordered balanced Laguerre-Sobolev type orthogonal polynomials

Date: 2006..05..??
Event: International congress COMPUMATG 2022
Venue: Universidad de Granma, Cuba

Abstract

In this contribution we consider a varying discrete Sobolev inner product involving the Laguerre polynomials. The aim of this contribution is twofold. First, we obtain algebraic properties and analytic properties for the polynomial sequence which is orthogonal with respect to the discrete Sobolev inner product

\[ \langle f, g\rangle_n=\langle {\bf u}_\alpha, fg\rangle+ \sum_{j=1}^M \mu_{j,n} f^{(\nu_j)}(c_j) \overline{g^{(\nu_j)}(c_j)}, \] where \({\bf u}_\alpha\) is the linear functional associated to the Laguerre polynomial, \(\alpha, c_j\in \mathbb C\), \(\nu_j\in \mathbb N_0\), \(j=1, 2,...., M\) and \(\mu_{j,n}\) are sequences of numbers satisfying \[ \lim_{n\to \infty} n^{\beta_j} \mu_{j,n} =\eta_j, \quad with \quad \beta_j\in \mathbb C. \]

Among them, we present several connection formulas for such polynomials, the varying Sobolev-orthogonal polynomials, we obtain an hypergeometric representation as well for them and we obtain a second-order holonomic differential equation that has this polynomial sequence as eigenfunctions. We also obtain results related to the zeros of this polynomial sequence as well as discuss an electrostatic model of their zeros. For some of the presented expressions we explicitly obtain their coefficients.

Second, we study the asymptotics behavior of the varying Sobolev-orthogonal polynomials. In this way, we focus our attention on Mehler-Heine type formulae as they describe in detail the asymptotic behavior of these polynomials around the mass-points \(c_j\), just the points where we have located the perturbations of the standard inner product.

Moreover, we pay attention to the asymptotic behavior of the zeros of these varying Sobolev polynomials and some numerical experiments are shown.

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