TY - JOUR

T1 - Congruences for overpartitions with restricted odd differences

AU - Hirschhorn, Michael D.

AU - Sellers, James A.

PY - 2020/10/1

Y1 - 2020/10/1

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.

KW - Congruences

KW - Overpartitions

KW - Restricted odd differences

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U2 - 10.1007/s11139-019-00156-x

DO - 10.1007/s11139-019-00156-x

M3 - Article

AN - SCOPUS:85069639959

VL - 53

SP - 167

EP - 180

JO - Ramanujan Journal

JF - Ramanujan Journal

SN - 1382-4090

IS - 1

ER -