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

Ernest X. W. Xia

Colloquium Mathematicae, Tome 139 (2015), p. 91-105 / Harvested from The Polish Digital Mathematics Library

Résumé

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

Publié le : 2015-01-01

Zbl 1327.11074

EUDML-ID : urn:eudml:doc:286142

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     author = {Ernest X. W. Xia},
     title = {New infinite families of Ramanujan-type congruences modulo 9 for overpartition pairs},
     journal = {Colloquium Mathematicae},
     volume = {139},
     year = {2015},
     pages = {91-105},
     zbl = {1327.11074},
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Ernest X. W. Xia. New infinite families of Ramanujan-type congruences modulo 9 for overpartition pairs. Colloquium Mathematicae, Tome 139 (2015) pp. 91-105. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm140-1-8/