New infinite families of Ramanujan-type congruences modulo 9 for overpartition pairs

Volume 140 / 2015

Ernest X. W. Xia Colloquium Mathematicum 140 (2015), 91-105 MSC: Primary 11P83; Secondary 05A17. DOI: 10.4064/cm140-1-8


Let $\overline{ pp}(n)$ denote the number of overpartition pairs of $n$. Bringmann and Lovejoy (2008) proved that for $n\geq 0$, $\, \overline{ pp} (3n+2) \equiv 0\pmod 3$. They also proved that there are infinitely many Ramanujan-type congruences modulo every power of odd primes for $\overline{ pp}(n)$. Recently, Chen and Lin (2012) established some Ramanujan-type identities and explicit congruences for $\overline{pp}(n)$. Furthermore, they also constructed infinite families of congruences for $\overline{ pp}(n)$ modulo 3 and 5, and two congruence relations modulo 9. In this paper, we prove several new infinite families of congruences modulo 9 for $\overline{ pp}(n)$. For example, we find that for all integers $k,n\geq 0$, $ \overline{ pp}( 2^{6k}(48n+20) ) \equiv \overline{ pp}( 2^{6k }(384n+32)) \equiv \overline{pp}( 2^{3k}(48n+36))\equiv 0 \pmod 9. $


  • Ernest X. W. XiaDepartment of Mathematics
    Jiangsu University
    Zhenjiang, Jiangsu 212013, P.R. China

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