Evolutio producti infiniti (1-x)(1-xx)(1-x³)(1-x⁴)(1-x⁵) etc. in seriem simplicem

2012/02/01 by Leonhard Euler, Euler, Leonhard, Artur Diener +3
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #History and Overview (math.HO) #History and Theory of Mathematics #math.HO

paper · pdf · doi:10.48550/arxiv.1202.0246

arxiv created 2012/02/01 · openalex publication_date 2012/02/01 · arxiv updated 2012/02/02 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

This paper does exactly what the title says it does. It expands the given series to arrive at the familiar "pentagonal number" expansion, also known as the pentagonal number theorem, and recalls its application to partition numbers. The paper is translated from Euler's Latin original into German.

Related

• Expansion of the infinite product (1-x)(1-xx)(1-x³)(1-x⁴)(1-x⁵)(1-x⁶) etc. into a simple series
• Euler and the pentagonal number theorem
• On the remarkable properties of the pentagonal numbers
• Various analytic observations on combinations
• An observation on the sums of divisors