Appell polynomial sequences with respect to some differential operators

TL;DR

This paper investigates polynomial sequences related to a specific type of lowering operator, characterizes their properties, and proves the non-existence of orthogonal polynomial sequences that are also $ ext{Lambda}$-Appell under certain conditions.

Contribution

It introduces a new class of $ ext{Lambda}$-Appell polynomial sequences defined via a finite sum of lowering operators and establishes their properties and limitations.

Findings

Characterization of $ ext{Lambda}$-Appell polynomial sequences.

Proof of non-existence of orthogonal $ ext{Lambda}$-Appell polynomials for a specific operator.

Analysis of cubic decompositions and their $ ext{Lambda}$-Appell properties.

Abstract

We present a study of a specific kind of lowering operator, herein called $\Lambda$, which is defined as a finite sum of lowering operators, proving that this configuration can be altered, for instance, by the use of Stirling numbers. We characterize the polynomial sequences fulfilling an Appell relation with respect to $\Lambda$, and considering a concrete cubic decomposition of a simple Appell sequence, we prove that the polynomial component sequences are $\Lambda$-Appell, with $\Lambda$ defined as previously, although by a three term sum. Ultimately, we prove the non-existence of orthogonal polynomial sequences which are also $\Lambda$-Appell, when $\Lambda$ is the lowering operator $\Lambda=a_{0}D+a_{1}DxD+a_{2}\left(Dx\right)^2D$, where $a_{0}$, $a_{1}$ and $a_{2}$ are constants and $a_{2} \neq 0$. The case where $a_{2}=0$ and $a_{1} \neq 0$ is also naturally recaptured.