Arithmetic Properties of Overpartitions into Odd Parts

Hirschhorn, Michael D.; Sellers, James A.

doi:10.1007/s00026-006-0293-7

Cited by 44 publications

(61 citation statements)

References 12 publications

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“…The generating function for po(n) is given by Many mathematicians have extensively studied the arithmetic properties of po(n) and they have also established several Ramanujan-type congruences satisfied by po(n) (for example, one can see [4,7]). Let A (n) denote the number of -regular overpartitions of n. The generating function for A (n) is given by…”

Section: N=1mentioning

confidence: 99%

Congruences modulo 8 for $$(2,\, k)$$ ( 2 , k ) -regular overpartitions for odd $$k > 1$$ k > 1

Adiga

Naika

Ranganatha³

et al. 2017

Arab. J. Math.

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Section: N=1mentioning

confidence: 99%

Congruences modulo 8 for $$(2,\, k)$$ ( 2 , k ) -regular overpartitions for odd $$k > 1$$ k > 1

Adiga

Naika

Ranganatha³

et al. 2017

Arab. J. Math.

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“…By employing dissection formulas, Fortin, Jacob and Mathieu [7], Hirschhorn and Sellers [9] independently derived various Ramanujan-type congruences for p(n), such as…”

Section: Introductionmentioning

confidence: 99%

“…Employing the 2-dissection formulas of theta functions due to Ramanujan, Hirschhorn and Sellers [9], Chen and Xia [4] obtained a generating function of p(40n + 35) modulo 5. Using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams [1], they showed that…”

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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“…As the topic of overpartitions has already been examined rather thoroughly [3,4,5,6,7,8,10,11], we look to new constructions. One such construction is that of an overpartition pair of a positive integer n, defined by Lovejoy [9] as a pair of overpartitions wherein the sum of all listed parts is n. For example, the overpartition pairs of 2 are Lovejoy denoted the number of overpartition pairs of a positive integer n by pp(n), with pp(0) = 1 by definition.…”

Section: Introductionmentioning

confidence: 99%

“…For example, the following were proven in [7]. Also, Hirschhorn and Sellers [6] proved that p(n) satisfies congruences modulo non-powers of 2 by proving the following:…”

Section: Introductionmentioning

confidence: 99%

Some Arithmetic Properties of Overpartition k-Tuples

Keister¹,

Sellers²,

Vary³

2009

Integers

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Recently, Lovejoy introduced the construct of overpartition pairs which are a natural generalization of overpartitions. Here we generalize that idea to overpartition ktuples and prove several congruences related to them. We denote the number of overpartition k-tuples of a positive integer n by p k (n) and prove, for example, that for all n ≥ 0, p t−1 (tn + r) ≡ 0 (mod t) where t is prime and r is a quadratic nonresidue mod t.

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Arithmetic Properties of Overpartitions into Odd Parts

Cited by 44 publications

References 12 publications

Congruences modulo 8 for $$(2,\, k)$$ ( 2 , k ) -regular overpartitions for odd $$k > 1$$ k > 1

Congruences modulo 8 for $$(2,\, k)$$ ( 2 , k ) -regular overpartitions for odd $$k > 1$$ k > 1

Ramanujan-type congruences for overpartitions modulo 16

Some Arithmetic Properties of Overpartition k-Tuples

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