Spherical coefficients of slice regular functions

TL;DR

This paper introduces a method to compute spherical coefficients of quaternionic slice regular functions using derivatives, compares these coefficients with those of the slice derivative, and explores related differential equations with detailed proofs and examples.

Contribution

It provides a direct, effective way to compute spherical coefficients and analyzes their relation to derivatives, offering new insights into quaternionic slice regular functions.

Findings

Effective computation of spherical coefficients from derivatives

Differential equations relating coefficients of functions and their derivatives

Detailed examples and alternative proofs included

Abstract

Given a quaternionic slice regular function $f$, we give a direct and effective way to compute the coefficients of its spherical expansion at any point. Such coefficients are obtained in terms of spherical and slice derivatives of the function itself. Afterwards, we compare the coefficients of $f$ with those of its slice derivative $\partial_{c}f$ obtaining a countable family of differential equations satisfied by any slice regular function. The results are proved in all details and are accompanied to several examples. For some of the results, we also give alternative proofs.