On Askey's extension of Clausen's identity and its polynomial perturbation

TL;DR

This paper extends Clausen's identity to arbitrary natural numbers and introduces polynomial perturbations, providing explicit formulas for these generalized hypergeometric series and their characteristic polynomials.

Contribution

It generalizes Askey's extension of Clausen's identity to all natural m and introduces polynomial perturbations, expanding the class of hypergeometric identities.

Findings

Extended Clausen's identity to all natural m.

Derived explicit formulas for polynomial-perturbed hypergeometric series.

Identified the degree of characteristic polynomials as 2m+s.

Abstract

The celebrated Clausen's identity expresses the square of the Gauss hypergeometric series ${}_2F_{1}(a,b;a+b+1/2;x)$ as a single hypergeometric ${}_3F_2$ series. Goursat showed in 1883 that replacing $1/2$ by $m+1/2$ leads to a hypergeometric series for the square whenever $m$ is a positive integer. Askey found this series explicitly for $m=1$. The first goal of this paper is to extend this result by treating the case of any natural $m$. The ${}_3F_{2}$ series on the right-hand side is thereby replaced by its perturbation by an explicit characteristic polynomial of degree $2m$, i.e., its coefficients are multiplied by values of this polynomial at nonnegative integers. The second goal of this paper is to make one further step and replace the square of the Gauss function by its product with its perturbation by an arbitrary polynomial of degree $s\le{2m+1}$. We show that such product remains hypergeometric and find its explicit form in terms of a polynomial perturbation of the ${}_3F_2$ series. We present an explicit formula for the characteristic polynomial whose degree is shown to be $2m+s$.