The tri-pentagonal number theorem and related identities

TL;DR

This paper revisits Andrews' pentagonal number theorem, uncovers a hidden polynomial identity, and uses it to prove the theorem along with new related formulas and an infinite family of multiple series identities.

Contribution

It reveals a simple polynomial identity underlying Riese's automated proof and introduces new formulas and a family of multiple series identities related to the tri-pentagonal theorem.

Findings

Uncovered a hidden polynomial identity in Riese's proof

Proved Andrews' pentagonal number theorem using this identity

Established a new infinite family of multiple series identities

Abstract

I revisit an automated proof of Andrews' pentagonal number theorem found by Riese. I uncover a simple polynomial identity hidden behind his proof. I explain how to use this identity to prove Andrews' result along with a variety of new formulas of similar type. I reveal an interesting relation between the tri-pentagonal theorem and items (19), (20), (94), (98) on the celebrated Slater list. Finally, I establish a new infinite family of multiple series identities.