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In this paper, we consider the asymptotic behavior of the sequence of monic polynomials orthogonal with respect to the Sobolev inner product 〈p,q〉S=∫0∞p(x)q(x)dμ+Mp(m)(ζ)q(m)(ζ), where ζ < 0, M ⩾ 0 and dμ = e−xxαdx. We study the outer relative asymptotics of these polynomials with respect to the classical Laguerre polynomials, and we deduce a Mehler–Heine type formula and a Plancherel–Rotach type formula for the rescaled polynomials.

We study the sequence of monic polynomials orthogonal with respect to inner product $$\begin{eqnarray*}\langle p, q\rangle = \int \nolimits \nolimits_{0}^{\infty } p(x)q(x){e}^{- x} {x}^{\alpha } \hspace{0.167em} dx+ Mp(\zeta )q(\zeta )+ N{p}^{\prime } (\zeta ){q}^{\prime } (\zeta ),\end{eqnarray*}$$ where $\alpha \gt - 1$, $M\geq 0$, $N\geq 0$, $\zeta \lt 0$, and $p$ and $q$ are polynomials with real coefficients. We deduce some interlacing properties of their zeros and, by using standard methods, we find a second-order linear differential equation satisfied by the polynomials and discuss an electrostatic model of their zeros.

En este artículo se estudian algunas técnicas de interpolación polinomial e interpolación polinomial a trozos. Respecto a las primeras, se propone una demostración alternativa de su equivalencia. También se plantea una sencilla codificación de tales métodos en el software Matlab, usando comandos básicos y sin recurrir a rutinas de interpolación predefinidas por este software.Palabras claves: Interpolación polinomial, interpolación polinomial a trozos, esplines cúbicos.AbstractThis article explores some polynomial interpolation and also piecewise polynomial interpolation methods. With respect to the first one, it is suggested an alternative demonstration of its equivalence. Also is posed a simple encoding of such methods in Matlab software, by using basic commands,without resorting to interpolation routines predefined by this software.Keywords: Polynomial interpolation, piecewise polynomial interpolation, cubic splines.

In this paper, we study an electrostatic model for zeros of polynomials orthogonal with respect to inner product