Trinomial equation: the Hypergeometric way

Abstract

This paper is devoted to the analytical treatment of trinomial equations of the form \(y^n+y=x,\) where \(y\) is the unknown and \(x\in\mathbb{C}\) is a free parameter. It is well-known that, for degree \(n\geq 5,\) algebraic equations cannot be solved by radicals; nevertheless, roots are described in terms of univariate hypergeometric or elliptic functions. This classical piece of research was founded by Hermite, Kronecker, Birkeland, Mellin and Brioschi, and continued by many other Authors. The approach mostly adopted in recent and less recent papers on this subject (see [<a href="#1">1</a>,<a href="#2">2</a>] for example) requires the use of power series, following the seminal work of Lagrange [<a href="#3">3</a>]. Our intent is to revisit the trinomial equation solvers proposed by the Italian mathematician Davide Besso in the late nineteenth century, in consideration of the fact that, by exploiting computer algebra, these methods take on an applicative and not purely theoretical relevance.