On Glaisher's Partition Theorem

TL;DR

This paper extends Glaisher's partition theorem by generalizing functions related to Euler's theorem, proving a new identity for the case m=3, and providing a new series representation of Glaisher's product.

Contribution

It generalizes the function D(n) and proves a new partition identity for m=3, expanding the understanding of Glaisher's theorem.

Findings

Proved a new partition identity for m=3.

Generalized the function D(n) related to Glaisher's theorem.

Derived a new series representation of Glaisher's product.

Abstract

Glaisher's theorem states that the number of partitions of $n$ into parts which repeat at most $m-1$ times is equal to the number of partitions of $n$ into parts which are not divisible by $m$. The $m=2$ case is Euler's famous partition theorem. Recently, Andrews, Kumar, and Yee gave two new partition functions $C(n)$ and $D(n)$ related to Euler's theorem. Lin and Zang extended their result to Glaisher's theorem by generalizing $C(n)$. We generalize $D(n)$ and prove an analogous partition identity for the $m=3$ case. We also provide a new series equal to Glaisher's product both in the finite and infinite cases.