On a sequence of monogenic polynomials satisfying the Appell condition whose first term is a non-constant function

TL;DR

This paper constructs a sequence of monogenic polynomials satisfying the Appell condition with a non-constant initial term, exploring their properties and connection to Fueter's theorem in hypercomplex analysis.

Contribution

It introduces a new class of monogenic polynomial sequences with non-constant initial terms satisfying the Appell condition, extending classical results.

Findings

Sequence constructed with non-constant initial term

Sequence satisfies the Appell condition in hypercomplex setting

Connection to Fueter's theorem discussed

Abstract

In this paper we aim at constructing a sequence $\{\mathsf{M}_n^k(x)\}_{n\ge0}$ of $\mathbb R_{0,m}$-valued polynomials which are monogenic in $\mathbb R^{m+1}$ satisfying the Appell condition (i.e. the hypercomplex derivative of each polynomial in the sequence equals, up to a multiplicative constant, its preceding term) but whose first term $\mathsf{M}_0^k(x)=\mathbf{P}_k(\underline x)$ is a $\mathbb R_{0,m}$-valued homogeneous monogenic polynomial in $\mathbb R^m$ of degree $k$ and not a constant like in the classical case. The connection of this sequence with the so-called Fueter's theorem will also be discussed.