Polynomial perturbations of Euler's and Clausen's identities

TL;DR

This paper extends classical hypergeometric identities, specifically Clausen's identity, by introducing polynomial perturbations, and provides new proofs and operators to facilitate these generalizations.

Contribution

It introduces a novel polynomial perturbation extension of Clausen's identity and develops the Miller-Paris operators, simplifying proofs and expanding hypergeometric transformation theory.

Findings

Extended Clausen's identity via polynomial perturbations.

New simplified proofs relating to polynomial interpolation.

Introduction of Miller-Paris operators for hypergeometric transformations.

Abstract

A product of two hypergeometric series is generally not hypergeometric. However, there are a few cases when such product does reduce to a single hypergeometric series. The oldest result of this type, beyond the obvious $(1-x)^{a}(1-x)^{b}=(1-x)^{a+b}$, is Euler's transformation for the Gauss hypergeometric function ${}_2F_1$. Another important one is the celebrated Clausen's identity dated 1828 which expresses the square of a suitable ${}_2F_1$ function as a single ${}_3F_2$. By equating coefficients each product identity corresponds to a special type of summation theorem for terminating series. Over the last two decades Euler's transformations and many summation theorems have been extended by introducing additional parameter pairs differing by positive integers. This amounts to multiplication of the power series coefficients by values of a fixed polynomial at nonnegative integers. The main goal of this paper is to present an extension of Clausen's identity obtained by such polynomial perturbation. To this end, we first reconsider the polynomial perturbations of Euler's transformations found by Miller and Paris around 2010. We propose new, simplified proofs of their transformations relating them to polynomial interpolation and exhibiting various new forms of the characteristic polynomials. We further introduce the notion of the Miller-Paris operators which play a prominent role in the construction of the extended Clausen's identity.