English

An Explanation of Mellin's 1921 Paper

Algebraic Geometry 2021-10-04 v1

Abstract

In 1921 Mellin published a Comptes Rendu paper computing the principal root of the polynomial Zn+x1Zn1++xpZnp1Z^n + x_1Z^{n_1} + \cdots + x_pZ^{n_p} - 1 using hypergeometric functions of its coefficients x1,...,xp.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.

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Cite

@article{arxiv.2110.00281,
  title  = {An Explanation of Mellin's 1921 Paper},
  author = {Wayne Lawton},
  journal= {arXiv preprint arXiv:2110.00281},
  year   = {2021}
}