Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product

TL;DR

This paper proves the nonnegativity of coefficients in truncated series related to Euler's pentagonal number theorem and Jacobi's triple product, offering new combinatorial insights and alternative proofs.

Contribution

It provides a new combinatorial proof for the nonnegativity of coefficients and establishes an analogous result for a truncated Jacobi triple product series.

Findings

Confirmed nonnegativity of coefficients in truncated Jacobi series

Provided a new combinatorial proof for Euler's pentagonal number theorem

Discovered an analogous nonnegativity result for Jacobi's triple product

Abstract

Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobi's triple product identity in a different manner.