Multiple extensions of a finite Euler's pentagonal number theorem and the Lucas formulas

TL;DR

This paper explores multiple extensions of Euler's pentagonal number theorem and Lucas formulas, providing new identities and a combinatorial proof, thus advancing the understanding of these classical number theory results.

Contribution

It introduces novel multiple extensions of Euler's pentagonal number theorem and unifies Lucas formulas through a common extension, including a combinatorial proof.

Findings

New multiple extensions of Euler's pentagonal number theorem

Unified extension of Lucas formulas

Combinatorial proof of Lucas' formulas

Abstract

Motivated by the resemblance of a multivariate series identity and a finite analogue of Euler's pentagonal number theorem, we study multiple extensions of the latter formula. In a different direction we derive a common extension of this multivariate series identity and two formulas of Lucas. Finally we give a combinatorial proof of Lucas' formulas.