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Truncated Jacobi triple product series II: Li, Lin, and Wang’s nonnegativity conjecture

Part of: Additive number theory; partitions Basic hypergeometric functions Combinatorics Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  19 June 2025

Shane Chern  and
Chun Wang
Shane Chern
Affiliation:
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, Wien 1090, Austria (chenxiaohang92@gmail.com)
Chun Wang*
Affiliation:
Department of Mathematics, Shanghai Normal University, Shanghai 200234, P.R. China (wangchun@shnu.edu.cn)
*
*Corresponding author.
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Abstract

The investigation of truncated theta series was popularized by Andrews and Merca. In this article, we establish an explicit expression with nonnegative coefficients for the bivariate truncated Jacobi triple product series:

\begin{equation*}\frac{1}{(z,q/z,q;q)_\infty}\sum_{n=0}^{m}(-1)^n(1-z^{n+1})(1-(q/z)^{n+1})q^{\binom{n+1}{2}},\end{equation*}

which can be regarded as a companion to Wang and Yee’s truncation of the triple product identity. As applications, our result confirms a conjecture of Li, Lin, and Wang and implies a family of linear inequalities for a bi-parametric partition function. We also work on another truncated triple product series arising from the work of Xia, Yee, and Zhao and derive similar nonnegativity results and linear inequalities.

Keywords

Bailey pairJacobi triple product identitylinear inequalities for partition functionsnonnegativitytruncated theta series

MSC classification

Primary: 11P84: Partition identities; identities of Rogers-Ramanujan type
Secondary: 11F27: Theta series; Weil representation; theta correspondences 05A20: Combinatorial inequalities 33D90: Applications

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 15
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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Footnotes

‡

This article originally published with errors. Those errors have now been corrected and the article updated. A correction notice has also been published.

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