On Polynomials Orthogonal with Respect to an Inner Product Involving Higher-Order Differences: The Meixner Case

TL;DR

This paper studies polynomials orthogonal with respect to a Sobolev-type inner product involving higher-order differences in the Meixner case, deriving explicit formulas, recurrence relations, ladder operators, and asymptotic behavior.

Contribution

It introduces a new class of Meixner-type orthogonal polynomials with higher-order difference inner products, providing explicit representations and key structural properties.

Findings

Explicit polynomial representations derived

Second-order linear difference equation established

Recurrence relations of order 2j+3 obtained

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.