A generalization of Clausen's identity

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.