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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-1-5,
author = {William Y. C. Chen and Ernest X. W. Xia},
title = {Proof of a conjecture of Hirschhorn and Sellers on overpartitions},
journal = {Acta Arithmetica},
volume = {166},
year = {2014},
pages = {59-69},
zbl = {1302.11084},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-1-5}
}
William Y. C. Chen; Ernest X. W. Xia. Proof of a conjecture of Hirschhorn and Sellers on overpartitions. Acta Arithmetica, Tome 166 (2014) pp. 59-69. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-1-5/