Δ-Sobolev orthogonal polynomials of Meixner type: asymptotics and limit relation

Abstract

Let $\{Q_n(x)\}_n$ be the sequence of monic polynomials orthogonal with respect to the Sobolev-type inner product $$\bigl\langle (p(x),r(x)\bigr \rangle_S=\bigl\langle{\bold u}_0,p(x)r(x) \bigr\rangle+ \lambda\bigl\langle {\bold u}_1,(\Delta p)(x)(\Delta r)(x) \bigr\rangle,$$ where $\lambda\ge 0$, $(\Delta f)(x)=f(x+1)-f(x)$ denotes the forward difference operator and $({\bold u}_0,{\bold u}_1)$ is a $\Delta$-coherent pair of positive-definite linear functionals being ${\bold u}_1$ the Meixner linear functional. In this paper, relative asymptotics for the $\{Q_n(x)\}_n$ sequence with respect to Meixner polynomials on compact subsets of $\bbfC\setminus[0,+\infty)$ is obtained. This relative asymptotics is also given for the scaled polynomials. In both cases, we deduce the same asymptotics as we have for the self-$\Delta$-coherent pair, that is, when ${\bold u}_0={\bold u}_1$ is the Meixner linear functional. Furthermore, we establish a limit relation between these orthogonal polynomials and the Laguerre-Sobolev orthogonal polynomials which is analogous to the one existing between Meixner and Laguerre polynomials in the Askey scheme.