Sequentially-ordered Sobolev inner product and Laguerre-Sobolev polynomials

Abstract

We study the sequence of polynomials $\{S_n\}_{n\geq 0}$ that are orthogonal with respect to the general discrete Sobolev-type inner product $$ \langle f,g \rangle_{\mathsf{s}}=\!\int\! f(x) g(x)d\mu(x)+\sum_{j=1}^{N}\sum_{k=0}^{d_j}\lambda_{j,k} f^{(k)}(c_j)g^{(k)}(c_j), $$ where $\mu$ is a finite Borel measure whose support $\supp{\mu}$ is an infinite set of the real line, $\lambda_{j,k}\geq 0$, and the mass points $c_i$, $i=1,\dots,N$ are real values outside the interior of the convex hull of $\supp{\mu}$ ($c_i\in\RR\setminus\inter{\ch{\supp{\mu}}}$). Under some restriction of order in the discrete part of $\langle \cdot, \cdot \rangle_{\mathsf{s}}$, we prove that $S_n$ has at least $n-d^*$ zeros on $\inter{\ch{\supp{\mu}}}$, being $d^*$ the number of terms in the discrete part of $\langle \cdot, \cdot \rangle_{\mathsf{s}}$. Finally, we obtain the outer relative asymptotic for $\{S_n\}$ in the case that the measure $\mu$ is the classical Laguerre measure, and for each mass point, only one order derivative appears in the discrete part of $\langle \cdot, \cdot \rangle_{\mathsf{s}}$.