Ramanujan-type congruences for overpartitions modulo 5

Abstract

Abstract. Let p(n) denote the number of overpartitions of n. Recently, Fortin-JacobMathieu and Hirschhorn-Sellers independently obtained 2-, 3-and 4-dissections of the generating function for p(n) and derived a number of congruences for p(n) modulo 4, 8 and 64 including p(5n + 2) ≡ 0 (mod 4), p(4n + 3) ≡ 0 (mod 8) and p(8n + 7) ≡ 0 (mod 64). By employing dissection techniques, Yao and Xia obtained congruences for p(n) modulo 8, 16 and 32, such as p(48n + 26) ≡ 0 (mod 8), p(24n + 17) ≡ 0 (mod 16) and p(72n+69) ≡ 0 (mod 32). In this paper, we give a 16-dissection of the generating function for p(n) modulo 16 and we show that p(16n + 14) ≡ 0 (mod 16) for n ≥ 0. Moreover, by using the 2-adic expansion of the generating function of p(n) due to Mahlburg, we obtain that p(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, we get p(49n + 7) ≡ 0 (mod 16) and p(49n+14) ≡ 0 (mod 16) for n ≥ 0. We also find four congruence relations: p(4n) ≡ (−1) n p(n) (mod 16) for n ≥ 0, p(4n) ≡ (−1) n p(n) (mod 32) for n being not a square of an odd positive integer, p(4n) ≡ (−1) n p(n) (mod 64) for n ≡ 1, 2, 5 (mod 8) and p(4n) ≡ (−1) n p(n) (mod 128) for n ≡ 0 (mod 4).

Restricted k-color partitions

“…Hirschhorn and Sellers show p(9 α (27n + 18)) ≡ 0 (mod 3) in [10]. That p(40n + 35) ≡ 0 (mod 5) was conjectured by Hirschhorn and Sellers and proved by Chen and Xia; the more general 4 k (40n + 35) was proved by W. Y. C. Chen, Sun, Wang and Zhang very recently [4], in a paper which also includes many other congruences mod 5 for the overpartition function that can be combined with modulus 7 for c 7,1 .…”

Congruences modulo 9 and 27 for overpartitions

“…Lin [12] presented a new proof of Hirschhorn and Sellers' conjecture. Chen, Sun, Wang and Zhang [4] deduced various Ramanujan-type congruences modulo 5 for overpartitions. Hirschhorn and Sellers [9], and Lovejoy and Osburn [14] discovered several congruences modulo 3 forp(n).…”

New Ramanujan type congruences modulo 5 for overpartitions

“…If we extract the terms in which the power of q is congruent to 2 modulo 5 from (4.2), divide by q 2 , and replace q 5 by q, we obtain the following result due to Chen et al [8,Eq. 3.12].…”

“…If we extract those terms in which the power of q is even and replace q 2 by q, we find that ∞ n=0 p(8n)q n ≡ 1 ϕ(−q) 8 …”

“…The unsettled congruence relation is that for n ≥ 0, Congruence relation (1.8) was recently proved by several authors, see [6,8,14,18]. All their methods rely on the generating function of p(5n) modulo 5.…”