Mathematics > Number Theory
[Submitted on 21 Nov 2024]
Title:Proof of Merca's stronger conjecture on truncated Jacobi triple product series
View PDF HTML (experimental)Abstract:The truncated series start off with Andrews and Merca's truncated version of Euler's pentagonal number theorem in 2012. Moreover, Andrews--Merca and Guo--Zeng independently conjectured that the truncated Jacobi triple product series has nonnegative coefficients, which has been proved analytically by Mao and combinatorially by Yee. In 2021, Merca gave a stronger version for the truncated Jacobi triple product series(JTPS). Some very spcial cases of the conjecture have been proved by Ballantine--Feigon, Ding-Sun and Zhou. On the one hand, we consider the partial finite denomintor of the stronger truncated JTPS series with Residue theorem and prove that the coefficients of $ q^{n} $ in the series are positive when $ n $ greater than a finite constant. On the other hand, for the infinte denomintor, we deduce an asymptotic of nonmodular infinite products and therefore prove Merca's conjecture when $ n $ greater than a constant. We also show than when $ k $ is large enough, the infinte denomintor can be ignored and the Conjecture holds as a corollary.
Current browse context:
math.NT
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.