A conjecture of Merca on nonnegativity of theta series

Abstract

In this paper, we will study a conjecture of Merca on theta series, which gives a refinement of a conjecture of Andrews and Merca on truncated pentagonal number series. We first show refinements of two special cases of Merca's conjecture and then establish several nonnegativity results on theta series. As applications, we establish positivity results involving two celebrated partition statistics.