Proof of a conjecture of Hirschhorn and Sellers on overpartitions

William Y. C. Chen, and Ernest X. W. Xia. "Proof of a conjecture of Hirschhorn and Sellers on overpartitions." Acta Arithmetica 163.1 (2014): 59-69. <http://eudml.org/doc/286504>.

@article{WilliamY2014,
abstract = {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.},
author = {William Y. C. Chen, Ernest X. W. Xia},
journal = {Acta Arithmetica},
keywords = {overpartition; congruence; theta function; dissection formula; -parametrization},
language = {eng},
number = {1},
pages = {59-69},
title = {Proof of a conjecture of Hirschhorn and Sellers on overpartitions},
url = {http://eudml.org/doc/286504},
volume = {163},
year = {2014},
}

TY - JOUR
AU - William Y. C. Chen
AU - Ernest X. W. Xia
TI - Proof of a conjecture of Hirschhorn and Sellers on overpartitions
JO - Acta Arithmetica
PY - 2014
VL - 163
IS - 1
SP - 59
EP - 69
AB - 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.
LA - eng
KW - overpartition; congruence; theta function; dissection formula; -parametrization
UR - http://eudml.org/doc/286504
ER -