Two truncated identities of Gauss

TL;DR

This paper introduces new expansions for partial sums of Gauss' series and derives inequalities related to overpartition and partition functions, proposing further conjectures.

Contribution

It presents novel expansions for Gauss' series and establishes new inequalities for partition-related functions, with conjectures for future research.

Findings

New expansions for partial sums of Gauss' series

Derived inequalities for overpartition and partition functions

Proposed conjectures for variations of partition functions

Abstract

Two new expansions for partial sums of Gauss' triangular and square numbers series are given. As a consequence, we derive a family of inequalities for the overpartition function $\bar{p}(n)$ and for the partition function $p_1(n)$ counting the partitions of $n$ with distinct odd parts. Some further inequalities for variations of partition function are proposed as conjectures.