A generalization of Clausen's identity

TL;DR

This paper extends Clausen's identity to a broader class of hypergeometric functions, providing new solutions and generalizations that encompass dihedral cases and related identities.

Contribution

It introduces a generalized form of Clausen's identity applicable to all Gauss hypergeometric functions and their solutions, expanding the theoretical framework.

Findings

Generalized Clausen's identity for all Gauss hypergeometric functions

Solutions expressed in bivariate series reducing to 3F2 series

Extended Chaundy's identity without parameter restrictions

Abstract

The paper aims to generalize Clausen's identity to the square of any Gauss hypergeometric function. Accordingly, solutions of the related 3rd order linear differential equation are found in terms of certain bivariate series that can reduce to 3F2 series similar to those in Clausen's identity. The general contiguous variation of Clausen's identity is found. The related Chaundy's identity is generalized without any restriction on the parameters of Gauss hypergeometric function. The special case of dihedral Gauss hypergeometric functions is underscored.