On Taylor series and Kapteyn series of the first and second type

TL;DR

This paper explores the mathematical relationship between Taylor series and Kapteyn series, providing explicit formulas to convert between the two and illustrating the method with examples.

Contribution

It introduces explicit formulas linking Taylor and Kapteyn series coefficients, enhancing understanding of their interrelation.

Findings

Derived formulas for converting between series types

Provided illustrative examples demonstrating the conversion

Enhanced mathematical understanding of series representations

Abstract

We study the relation between the coefficients of Taylor series and Kapteyn series representing the same function. We compute explicit formulas for expressing one in terms of the other and give examples to illustrate our method.