Translator Disclaimer
December 2020 Ramanujan-type congruences for overpartitions modulo $3$
Li Zhang
Rocky Mountain J. Math. 50(6): 2257-2264 (December 2020). DOI: 10.1216/rmj.2020.50.2257

Abstract

Let p̄(n) denote the number of overpartitions of n. We show that p̄(3n)p̄(93n)(mod3) for n0 and p̄(3n)(1)np̄(163n)(mod3) for n0 by using a relation of the generating function of p̄(3n) modulo 3 and elementary dissection manipulations. Furthermore, by studying a 4-dissection formula of the generating function of p̄(3n) modulo 3 and iterating the above two congruence relations, we derive that p̄(16α9β(72n+51))0(mod3) for α,β,n0. Moreover, applying the fact that ϕ(q)5 is a Hecke eigenform in M52(Γ˜0(4)), we obtain an infinite family of congruences p̄(16α9β32n)0(mod3), where α,β0 and is a prime such that 1(mod3) and (n)=1. In this way, we find various Ramanujan-type congruences for p̄(n) modulo 3 such as p̄(1029n+441)0(mod3), p̄(1029n+735)0(mod3) and p̄(1029n+882)0(mod3) for n0.

Citation

Download Citation

Li Zhang. "Ramanujan-type congruences for overpartitions modulo $3$." Rocky Mountain J. Math. 50 (6) 2257 - 2264, December 2020. https://doi.org/10.1216/rmj.2020.50.2257

Information

Received: 2 May 2020; Revised: 23 June 2020; Accepted: 26 June 2020; Published: December 2020
First available in Project Euclid: 5 January 2021

Digital Object Identifier: 10.1216/rmj.2020.50.2257

Subjects:
Primary: 05A17, 11P83

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

JOURNAL ARTICLE
8 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.50 • No. 6 • December 2020
Back to Top