On analytic properties of Meixner-Sobolev orthogonal polynomials of higher order difference operators

Abstract

In this contribution we consider sequences of monic polynomials orthogonal with respect to Sobolev-type inner product $$\left\langle f,g\right\rangle= \langle {\bf u}^{\tt M},fg\rangle+\lambda \mathscr T^j f (\alpha)\mathscr T^{j}g(\alpha),$$ where ${\bf u}^{\tt M}$ is the Meixner linear operator, $\lambda\in\mathbb{R}_{+}$, $j\in\mathbb{N}$, $\alpha \leq 0$, and $\mathscr T$ is the forward difference operator $\Delta$, or the backward difference operator $\nabla$. We derive an explicit representation for these polynomials. The ladder operators associated with these polynomials are obtained, and the linear difference equation of second order is also given. In addition, for these polynomials we derive a $(2j+3)$-term recurrence relation. Finally, we find the Mehler-Heine type formula for the $\alpha\le 0$ case