Ramanujan-type Congruences for Overpartitions Modulo 16
William Y.C. Chen, Qing-Hu Hou, Lisa H. Sun and Li Zhang
Abstract: Let (n) denote the number of overpartitions of n. Recently, Fortin-Jacob-Mathieu and Hirschhorn-Sellers independently obtained 2-, 3- and 4-dissections of the generating function for (n) and derived a number of congruences for (n) modulo 4, 8 and 64 including (8n+7) ≡ 0 (mod 64), for n ≥ 0. In this paper, we give a 16-dissection of the generating function for (n) modulo 16 and show that (16n+14) ≡ 0 (mod 16) for n ≥ 0. Moreover, using the 2-adic expansion of the generating function for (n) according to Mahlburg, we obtain that (l^{2}n + rl) ≡ 0 (mod 16), where n ≥ 0, l ≡ -1 (mod 8) is an odd prime and r is a positive integer with l r. In particular, for l = 7 and n ≥ 0, we get (49n+7) ≡ 0 (mod 16) and (49n+14) ≡ 0 (mod 16). We also find four congruence relations: (4n) ≡ (-1)^{n}(n) (mod 16) for n ≥ 0, (4n) ≡ (-1)^{n}(n) (mod 32) where n is not a square of an odd positive integer, (4n) ≡ (-1)^{n}(n) (mod 64) for n 1, 2, 5 (mod 8) and (4n) ≡ (-1)^{n}(n) (mod 128) for n ≡ 0 (mod 4). AMS Classification: 05A17, 11P83 Keywords: overpartition, Ramanujan-type congruence, 2-adic expansion, dissection formula Download: pdf |