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 $\langle \cdot, \cdot \rangle_{ \mathsf{ s}}$.
Categories: math.CA math.CV