A generalization of Franklin's partition identity and a Beck-type companion identity

TL;DR

This paper generalizes Franklin's partition identity, extending Euler's classic result, and proves a Beck-type companion identity, broadening the understanding of partition identities in combinatorics.

Contribution

It introduces a new generalization of Franklin's identity that encompasses previous results and establishes a related Beck-type companion identity.

Findings

Generalized Franklin's partition identity

Proved a Beck-type companion identity

Unified previous partition identities

Abstract

Euler's classic partition identity states that the number of partitions of $n$ into odd parts equals the number of partitions of $n$ into distinct parts. We develop a new generalization of this identity, which yields a previous generalization of Franklin as a special case, and prove an accompanying Beck-type companion identity.