A general positivity result on coefficients of certain $q$-series

TL;DR

This paper proves a broad positivity result for coefficients of specific $q$-series, unifying and extending classical identities, and confirms two conjectures related to positivity in partition theory.

Contribution

It introduces a general positivity theorem for $q$-series coefficients that refines multiple classical identities and verifies two conjectures by Merca.

Findings

Established a unified positivity result for certain $q$-series coefficients

Refined classical identities like the pentagonal number series and Gauss' identities

Confirmed two positivity conjectures of Merca

Abstract

Based on a classical result on partitions of an integer into a finite set of positive integers, we establish a general positivity result on coefficients of certain $q$-series which uniformly refines the positivity of truncated pentagonal number series, truncated Gauss' identities and some special cases of truncated Jacobi triple product identity. As an application, we prove two positivity conjectures due to Merca.