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

Ernest X. W. Xia¹

¹ Department of Mathematics Jiangsu University Zhenjiang, Jiangsu 212013, P.R. China

Colloquium Mathematicum, Tome 140 (2015) no. 1, pp. 91-105.

Résumé

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. $

Détail

DOI : 10.4064/cm140-1-8

Keywords: overline denote number overpartition pairs bringmann lovejoy proved geq overline equiv pmod proved there infinitely many ramanujan type congruences modulo every power odd primes overline recently chen lin established ramanujan type identities explicit congruences overline furthermore constructed infinite families congruences overline modulo nbsp nbsp congruence relations modulo nbsp paper prove several infinite families congruences modulo nbsp overline example integers geq overline equiv overline equiv overline equiv pmod