A note on some identities involving special functions from the hypergeometric solution of algebraic equations

TL;DR

This paper derives new reduction formulas for hypergeometric functions from algebraic solutions of specific polynomial equations, enabling simpler expressions and calculations of certain integrals in elementary functions.

Contribution

It introduces novel reduction formulas for hypergeometric functions based on algebraic solutions of polynomial equations and their derivatives.

Findings

New reduction formulas for hypergeometric functions

Simplified expressions for certain infinite integrals

Connections between algebraic solutions and special functions

Abstract

From the algebraic solution of $x^{n}-x+t=0$ for $n=2,3,4$ and the corresponding solution in terms of hypergeometric functions, we obtain a set of reduction formulas for hypergeometric functions. By differentiation and integration of these results, and applying other known reduction formulas of hypergeometric functions, we derive new reduction formulas of special functions as well as the calculation of some infinite integrals in terms of elementary functions.