Congruences for overpartitions with restricted odd differences

Michael D. Hirschhorn, James A. Sellers

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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)
Pages (from-to)167-180
Number of pages14
JournalRamanujan Journal
Volume53
Issue number1
DOIs
StatePublished - Oct 1 2020
Externally publishedYes

Keywords

  • Congruences
  • Overpartitions
  • Restricted odd differences

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