On Glaisher's Partition Theorem

George E. Andrews; Aritram Dhar

Paper

On Glaisher's Partition Theorem

Abstract

Glaisher's theorem states that the number of partitions of

into parts which repeat at most

times is equal to the number of partitions of

into parts which are not divisible by

. The

case is Euler's famous partition theorem. Recently, Andrews, Kumar, and Yee gave two new partition functions

and

related to Euler's theorem. Lin and Zhang extended their result to Glaisher's theorem by generalizing

. We generalize

, prove an analogous partition identity for the

case, and show that the general case is an example of an almost partition identity. We also provide a new series equal to Glaisher's product both in the finite and infinite cases.