Multiplicative properties of the number of $k$-regular partitions

TL;DR

This paper explores multiplicative properties of $k$-regular partitions, identifying maximum points of a generating function and extending classical results to derive new inequalities for these partitions.

Contribution

It introduces a multiplicative extension of the generating function for $k$-regular partitions and characterizes its maximum, extending Lehmer's classical result.

Findings

Maximum of the extended generating function is at a small, explicitly described set of partitions.

Derived a new inequality for the generating function of $k$-regular partitions.

Extended classical results to analyze multiplicative properties of $k$-regular partitions.

Abstract

In a previous paper of the second author with K. Ono, surprising multiplicative properties of the partition function were presented. Here, we deal with $k$-regular partitions. Extending the generating function for $k$-regular partitions multiplicatively to a function on $k$-regular partitions, we show that it takes its maximum at an explicitly described small set of partitions, and can thus easily be computed. The basis for this is an extension of a classical result of Lehmer, from which an inequality for the generating function for $k$-regular partitions is deduced which seems not to have been noticed before.