Composition-theoretic series in partition theory

TL;DR

This paper introduces a novel composition-theoretic approach to partition enumeration, utilizing sums over compositions into k-gonal numbers and employing Ramanujan's theta functions, with applications to lacunary q-series and Dirichlet series.

Contribution

It presents a new method connecting compositions and partition functions through theta functions, expanding the analytical tools in partition theory.

Findings

Expressed partition functions as sums over compositions into k-gonal parts

Applied theta functions to derive new identities in partition enumeration

Introduced a new class of composition-theoretic Dirichlet series

Abstract

We use sums over integer compositions analogous to generating functions in partition theory, to express certain partition enumeration functions as sums over compositions into parts that are $k$-gonal numbers; our proofs employ Ramanujan's theta functions. We explore applications to lacunary $q$-series, and to a new class of composition-theoretic Dirichlet series.