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Proof of a conjecture of Hirschhorn and Sellers on overpartitions

100%

Chen W., Xia E.

Acta Arithmetica

2014

tom 163

nr 1

59-69

Let p̅(n) denote the number of overpartitions of n. It was conjectured by Hirschhorn and Sellers that p̅(40n+35) ≡ 0 (mod 40) for n ≥ 0. Employing 2-dissection formulas of theta functions due to Ramanujan, and Hirschhorn and Sellers, we obtain a generating function for p̅(40n+35) modulo 5. Using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we prove the congruence p̅(40n+35) ≡ 0 (mod 5) for n ≥ 0. Combining this congruence and the congruence p̅(4n+3) ≡ 0 (mod 8) for n ≥ 0 obtained by Hirschhorn and Sellers, and Fortin, Jacob and Mathieu, we confirm the conjecture of Hirschhorn and Sellers.

Arithmetic properties of overpartition pairs

100%

Chen W., Lin B.

Acta Arithmetica

2012

tom 151

nr 3

263-277

Ograniczanie wyników

2 Acta Arithmetica

2 Chen W.

1 Lin B.

1 Xia E.

1 2014

1 2012

Wyniki wyszukiwania

Proof of a conjecture of Hirschhorn and Sellers on overpartitions

Arithmetic properties of overpartition pairs