On the expansion of the power of any polynomial 1+x+x²+x³+x⁴+etc

2005/05/20 by Leonhard Euler, Euler, Leonhard
Mathematics · #01A50 #05A10 #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #History and Overview (math.HO) #History and Theory of Mathematics #Mathematics and Applications #math.CO #math.HO #msc:01A50 #msc:05A10

paper · pdf · doi:10.48550/arxiv.math/0505425

8 pages. Seems to be first English translation of Euler's Latin original ``De evolutione potestatis polynomialis cuiuscunque (1+x+x^2+x^3+x^4+etc.)^n, Nova Acta Academiae Scientarum Imperialis Petropolitinae 12 (1801), 47-57

arxiv created 2005/05/20 · openalex publication_date 2005/05/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, Euler gives the general trionomial coefficient as a sum of the binomial coefficients, the general quadrinomial coefficient as a sum of the binomial and trinomial coefficients, the general quintonomial coefficient as a sum of the binomial and quadrinomial coefficients, and gives a general determination of the coefficients of the expansion of any polynomial (1+x+x²+...+x^m)ⁿ as a sum of the coefficients of lower degree polynomial coefficients.

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