Relations for partitions with distinct even parts except the largest part which is even

TL;DR

This paper establishes new q-series identities linking 4-regular partitions with partitions having distinct even parts, introduces three novel partition functions, and explores their congruences.

Contribution

It introduces three new partition functions with specific even part restrictions and proves identities connecting them to 4-regular partitions, along with congruence results.

Findings

New q-series identities connecting 4-regular and special partitions.

Definition of three new partition functions with even part restrictions.

Congruence properties for the new partition functions.

Abstract

In this paper, we prove some new \(q\)-series identities connecting \(4\)-regular partitions and partitions with distinct even parts with largest part being odd. We also define three new partition functions with distinct even parts except the largest part which is even, and prove identities connecting the three partitions with \(4\)-regular partitions. Moreover, we also offer some congruence for the three newly defined partitions.