Hecke-type series involving infinite products

TL;DR

This paper investigates Hecke-type series involving infinite products, establishing new series, their truncated versions, and deriving inequalities for partition functions, building on prior foundational work.

Contribution

It introduces new Hecke-type series involving infinite products and derives inequalities for partition functions, expanding the mathematical understanding of these series.

Findings

New Hecke-type series involving infinite products are established.

Truncated versions of these series are derived.

Inequalities for certain partition functions are obtained.

Abstract

Since the study by Jacobi and Hecke, Hecke-type series have received extensive attention. Especially, Hecke-type series involving infinite products have attracted broad interest among many mathematicians including Kac, Peterson, Andrews, Bressoud and Liu. Motivated by the works of these people, we study Hecke-type series involving infinite products. In particular, we establish some Hecke-type series involving infinite products and then obtain the truncated versions of these series as well as some other known series of the same type. As consequences, three families of inequalities for certain partition functions are also presented. Our proofs heavily rely on a formula from the work of Zhi-Guo Liu.