Congruences of multipartition functions modulo powers of primes

Chen, William Y. C.; Du, Daniel K.; Hou, Qing-Hu; Sun, Lisa H.

doi:10.1007/s11139-013-9485-z

Cited by 9 publications

(7 citation statements)

References 29 publications

(68 reference statements)

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“…To make this paper self-contained, we recall some definitions and notation on half-integral weight modular forms. For more details, see [3,14,[19][20][21].…”

Section: Preliminariesmentioning

confidence: 99%

“…Based on the Hecke operator on φ(q) 3 and the fact that φ(q) 3 is a Hecke eigenform in M3 2 (Γ 0 (4)), we obtain a family of congruences for overpartitions modulo 5.…”

Section: Introductionmentioning

confidence: 99%

“…In this section, we give proofs of Theorem 1.1 and Theorem 1.2 by using a relation of the generating function of p(5n) modulo 5 and the 2-adic expansion (1.2) of p(n).Proof of Theorem 1.1. Recall the following 2-dissection formula for φ(q), φ(q) = φ(q 4 ) + 2qψ(q 8 ),(3…”

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confidence: 99%

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Ramanujan-type congruences for overpartitions modulo 5

Chen

Sun

Wang

et al. 2015

Journal of Number Theory

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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).

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“…To make this paper self-contained, we recall some definitions and notation on half-integral weight modular forms. For more details, see [3,14,[19][20][21].…”

Section: Preliminariesmentioning

confidence: 99%

“…Based on the Hecke operator on φ(q) 3 and the fact that φ(q) 3 is a Hecke eigenform in M3 2 (Γ 0 (4)), we obtain a family of congruences for overpartitions modulo 5.…”

Section: Introductionmentioning

confidence: 99%

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Ramanujan-type congruences for overpartitions modulo 5

Chen

Sun

Wang

et al. 2015

Journal of Number Theory

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“…Later on, Chen et al [5,Eq. (1.17)] proved the following congruence modulo 25 for p −2 (n) via modular forms: p −2 (25n + 23) ≡ 0 (mod 25).…”

Section: Introductionmentioning

confidence: 99%

Congruences modulo powers of 5 for k-colored partitions

Tang

2018

Journal of Number Theory

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Abstract. Let p −k (n) enumerate the number of k-colored partitions of n. In this paper, we establish some infinite families of congruences modulo 25 for k-colored partitions. Furthermore, we prove some infinite families of Ramanujan-type congruences modulo powers of 5 for p −k (n) with k = 2, 6, and 7. For example, for all integers n ≥ 0 and α ≥ 1, we prove thatand p −2 5 2α n + 11 × 5 2α + 1 12 ≡ 0 (mod 5 α+1 ).

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“…We set p(0) = 1 and p(n) = 0 if n < 0. For example, there are eight overpartitions of 3 3,3, 2 + 1,2 + 1, 2 +1,2 +1, 1 + 1 + 1,1 + 1 + 1.…”

Section: Introductionmentioning

confidence: 99%

Ramanujan-type congruences for overpartitions modulo 16

et al. 2015

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Abstract. Let p(n) denote the number of overpartitions of n. Hirschhorn and Sellers showed that p(4n + 3) ≡ 0 (mod 8) for n ≥ 0. They also conjectured that p(40n + 35) ≡ 0 (mod 40) for n ≥ 0. Chen and Xia proved this conjecture by using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams. In this paper, we show that p(5n) ≡ (−1) n p(4 · 5n) (mod 5) for n ≥ 0 and p(n) ≡ (−1) n p(4n) (mod 8) for n ≥ 0 by using the relation of the generating function of p(5n) modulo 5 found by Treneer and the 2-adic expansion of the generating function of p(n) due to Mahlburg. As a consequence, we deduce that p(4 k (40n + 35)) ≡ 0 (mod 40) for n, k ≥ 0. Furthermore, applying the Hecke operator on φ(q) 3 and the fact that φ(q) 3 is a Hecke eigenform, we obtain an infinite family of congrences p(4 k · 5ℓ 2 n) ≡ 0 (mod 5), where k ≥ 0 and ℓ is a prime such that ℓ ≡ 3 (mod 5) and −n ℓ = −1. Moreover, we show that p(5 2 n) ≡ p(5 4 n) (mod 5) for n ≥ 0. So we are led to the congruences p 4 k 5 2i+3 (5n ± 1) ≡ 0 (mod 5) for n, k, i ≥ 0. In this way, we obtain various Ramanujan-type congruences for p(n) modulo 5 such as p(45(3n + 1)) ≡ 0 (mod 5) and p(125(5n ± 1)) ≡ 0 (mod 5) for n ≥ 0.

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Congruences of multipartition functions modulo powers of primes

Cited by 9 publications

References 29 publications

Ramanujan-type congruences for overpartitions modulo 5

Ramanujan-type congruences for overpartitions modulo 5

Congruences modulo powers of 5 for k-colored partitions

Ramanujan-type congruences for overpartitions modulo 16

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