Trigonometric identities: from Hermite via Meijer, N{\o}rlund and Braaksma to Chu and Johnson and beyond

TL;DR

This paper explores classical and modern trigonometric identities, filling historical gaps, and extending recent results by integrating identities from Meijer, N{ o}rlund, and Braaksma with hypergeometric function theory.

Contribution

It uncovers and systematizes a wide range of trigonometric identities, connecting historical results with contemporary generalizations and providing new extensions and explicit examples.

Findings

Identifies and presents identities by Meijer, N{ o}rlund, and Braaksma from 1940-1962.

Extends Chu's and Johnson's identities using complex analysis and sum manipulations.

Provides numerous explicit examples illustrating the unified identities.

Abstract

Known already to the ancient Greeks, today trigonometric identities come in a large variety of tastes and flavours. In this large family there is a subfamily of interpolation-like identities discovered by Hermite and revived rather recently in two independent papers, one by Wenchang Chu and the other by Warren Johnson exploring various forms and generalizations of Hermite's results. The goal of this work inspired by these two articles is twofold. The first goal is to fill a gap in the references from the above papers and exhibit various trigonometric identities discovered by Meijer, N{\o}rlund and Braaksma between 1940 and 1962 in the context of analytic continuation of Mellin-Barnes integrals and relations between different solutions of the generalized hypergeometric differential equation. Our second goal is to present some extensions of Chu's and Johnson's results by combining them with the ideas of Meijer and Braaksma adding certain sum manipulations and facts from the complex analysis. We unify and systematize various known and new identities and illustrate our results with numerous explicit examples.