On Euler's Solution of the simple Difference Equation

TL;DR

This paper examines Euler's method for solving simple difference equations and derives values of the Riemann zeta function at positive integers using Euler's original ideas.

Contribution

It provides a detailed analysis of Euler's solution to difference equations and offers a derivation of zeta function values based on Euler's concepts.

Findings

Euler's solution to the difference equation is elucidated.

A derivation of the Riemann zeta function at positive integers is presented.

The paper connects Euler's methods to modern understanding of series and special functions.

Abstract

In this note we will discuss Euler's solution of the simple difference equation that he gave in his paper{\it ``De serierum determinatione seu nova methodus inveniendi terminos generales serierum"} \cite{E189} (E189:``On the determination of series or a new method of finding the general terms of series") and also present a derivation for the values of the Riemann $\zeta$-function at positive integer numbers based on Euler's ideas.