A Note on DeMoivre's Quintic Equation

TL;DR

This paper presents a radical solution to a specific form of the quintic equation and uses these solutions to evaluate a hypergeometric series for various arguments.

Contribution

It provides a radical solution to DeMoivre's quintic and applies it to sum hypergeometric series, offering new analytical tools.

Findings

Radical solutions for the specific quintic form are derived.

Hypergeometric series are summed explicitly for several arguments.

The approach links algebraic solutions with special functions.

Abstract

The quintic equation with real coefficients $$x^5+5ax^3+5a^2x+b=0$$ is solved in terms of radicals and the results used to sum a hypergeometric series for several arguments.