Congruences for overpartitions with restricted odd differences

Michael D. Hirschhorn, James A. Sellers

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

In a recent work, Bringmann, Dousse, Lovejoy, and Mahlburg defined the function t¯ (n) to be the number of overpartitions of weight n where (i) the difference between two successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd, then it is overlined. In their work, they proved that t¯ (n) satisfies an elegant congruence modulo 3, namely, for n≥ 1 , [Equation not available: see fulltext.]In this work, using elementary tools for manipulating generating functions, we prove that t¯ satisfies a corresponding parity result. We prove that, for all n≥ 1 , t¯(2n)≡{1(mod2)ifn=(3k+1)2for some integerk,0(mod2)otherwise.We also provide a truly elementary proof of the mod 3 characterization of Bringmann et al., as well as a number of additional congruences satisfied by t¯ (n) for various moduli.

Original languageEnglish (US)
JournalRamanujan Journal
DOIs
StatePublished - Jan 1 2019

Fingerprint

Congruence
Odd
Parity
Generating Function
Modulo
Modulus

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

Hirschhorn, Michael D. ; Sellers, James A. / Congruences for overpartitions with restricted odd differences. In: Ramanujan Journal. 2019.
@article{56d2e27e038b425c8a0593d2b34a7454,
title = "Congruences for overpartitions with restricted odd differences",
abstract = "In a recent work, Bringmann, Dousse, Lovejoy, and Mahlburg defined the function t¯ (n) to be the number of overpartitions of weight n where (i) the difference between two successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd, then it is overlined. In their work, they proved that t¯ (n) satisfies an elegant congruence modulo 3, namely, for n≥ 1 , [Equation not available: see fulltext.]In this work, using elementary tools for manipulating generating functions, we prove that t¯ satisfies a corresponding parity result. We prove that, for all n≥ 1 , t¯(2n)≡{1(mod2)ifn=(3k+1)2for some integerk,0(mod2)otherwise.We also provide a truly elementary proof of the mod 3 characterization of Bringmann et al., as well as a number of additional congruences satisfied by t¯ (n) for various moduli.",
author = "Hirschhorn, {Michael D.} and Sellers, {James A.}",
year = "2019",
month = "1",
day = "1",
doi = "10.1007/s11139-019-00156-x",
language = "English (US)",
journal = "Ramanujan Journal",
issn = "1382-4090",
publisher = "Springer Netherlands",

}

Congruences for overpartitions with restricted odd differences. / Hirschhorn, Michael D.; Sellers, James A.

In: Ramanujan Journal, 01.01.2019.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Congruences for overpartitions with restricted odd differences

AU - Hirschhorn, Michael D.

AU - Sellers, James A.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - In a recent work, Bringmann, Dousse, Lovejoy, and Mahlburg defined the function t¯ (n) to be the number of overpartitions of weight n where (i) the difference between two successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd, then it is overlined. In their work, they proved that t¯ (n) satisfies an elegant congruence modulo 3, namely, for n≥ 1 , [Equation not available: see fulltext.]In this work, using elementary tools for manipulating generating functions, we prove that t¯ satisfies a corresponding parity result. We prove that, for all n≥ 1 , t¯(2n)≡{1(mod2)ifn=(3k+1)2for some integerk,0(mod2)otherwise.We also provide a truly elementary proof of the mod 3 characterization of Bringmann et al., as well as a number of additional congruences satisfied by t¯ (n) for various moduli.

AB - In a recent work, Bringmann, Dousse, Lovejoy, and Mahlburg defined the function t¯ (n) to be the number of overpartitions of weight n where (i) the difference between two successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd, then it is overlined. In their work, they proved that t¯ (n) satisfies an elegant congruence modulo 3, namely, for n≥ 1 , [Equation not available: see fulltext.]In this work, using elementary tools for manipulating generating functions, we prove that t¯ satisfies a corresponding parity result. We prove that, for all n≥ 1 , t¯(2n)≡{1(mod2)ifn=(3k+1)2for some integerk,0(mod2)otherwise.We also provide a truly elementary proof of the mod 3 characterization of Bringmann et al., as well as a number of additional congruences satisfied by t¯ (n) for various moduli.

UR - http://www.scopus.com/inward/record.url?scp=85069639959&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85069639959&partnerID=8YFLogxK

U2 - 10.1007/s11139-019-00156-x

DO - 10.1007/s11139-019-00156-x

M3 - Article

AN - SCOPUS:85069639959

JO - Ramanujan Journal

JF - Ramanujan Journal

SN - 1382-4090

ER -