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

Ernest X. W. Xia. "New infinite families of Ramanujan-type congruences modulo 9 for overpartition pairs." Colloquium Mathematicae 140.1 (2015): 91-105. <http://eudml.org/doc/286142>.

@article{ErnestX2015,
abstract = {Let $\overline\{pp\}(n)$ denote the number of overpartition pairs of n. Bringmann and Lovejoy (2008) proved that for n ≥ 0, $\overline\{pp\}(3n+2) ≡ 0 (mod 3)$. 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 ≥ 0, $\overline\{pp\}(2^\{6k\}(48n+20)) ≡ \overline\{pp\}(2^\{6k\}(384n+32)) ≡ \overline\{pp\}(2^\{3k\}(48n+36)) ≡ 0 (mod 9)$.},
author = {Ernest X. W. Xia},
journal = {Colloquium Mathematicae},
keywords = {overpartition pair; congruence; generating function; theta function},
language = {eng},
number = {1},
pages = {91-105},
title = {New infinite families of Ramanujan-type congruences modulo 9 for overpartition pairs},
url = {http://eudml.org/doc/286142},
volume = {140},
year = {2015},
}

TY - JOUR
AU - Ernest X. W. Xia
TI - New infinite families of Ramanujan-type congruences modulo 9 for overpartition pairs
JO - Colloquium Mathematicae
PY - 2015
VL - 140
IS - 1
SP - 91
EP - 105
AB - Let $\overline{pp}(n)$ denote the number of overpartition pairs of n. Bringmann and Lovejoy (2008) proved that for n ≥ 0, $\overline{pp}(3n+2) ≡ 0 (mod 3)$. 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 ≥ 0, $\overline{pp}(2^{6k}(48n+20)) ≡ \overline{pp}(2^{6k}(384n+32)) ≡ \overline{pp}(2^{3k}(48n+36)) ≡ 0 (mod 9)$.
LA - eng
KW - overpartition pair; congruence; generating function; theta function
UR - http://eudml.org/doc/286142
ER -