An Explanation of Mellin's 1921 Paper

Abstract

In 1921 Mellin published a Comptes Rendu paper computing the principal root of the polynomial $Z^n+x_1{Z}^{n_1}+\cdots +x_p{Z}^{n_p}-1$ using hypergeometric functions of its coefficients $x_1,...,x_p.$ He used an integral transform nowadays bearing his name. Slightly over three pages, the paper is written in French in a terse style befitting the language. Unable to find an elementary explanation on the web or in a texbook, we wrote this expository article to make Mellin's landmark result accessible to interested people who are not experts in hypergeometric functions and complex analysis.