A poset version of Ramanujan results on Eulerian numbers and zeta values

TL;DR

This paper generalizes Ramanujan's results on Eulerian numbers and zeta values using poset operads and order polytopes, revealing new algebraic structures and connections to zeta value independence.

Contribution

It introduces a poset-based framework to extend Ramanujan's identities and provides new proofs and insights into the algebraic and combinatorial properties of Eulerian numbers and zeta values.

Findings

Generalized Ramanujan's zeta value identities

Provided new proofs of Eulerian number properties

Linked zeta value properties to algebraic independence

Abstract

We explore the operad of finite posets and its algebras. We use order polytopes to investigate the combinatorial properties of zeta values. By generalizing a family of zeta value identities, we demonstrate the applicability of this approach. In addition, we offer new proofs of some of Ramanujan's results on the properties of Eulerian numbers, interpreting his work as dealing with series inheriting the algebraic structure of disjoint unions of points. Finally, we establish a connection between our findings and the linear independence of zeta values.