Abstract: Let (n)
denote the number of overpartitions of n. It was conjectured by
Hirschhorn and Sellers that (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
(40n + 35) modulo 5. Using the (p, k)-parametrization
of theta functions given by Alaca, Alaca and Williams, we prove the congruence
(40n + 35) ≡ 0 (mod 5).
Combining this congruence and the congruence
(4n + 3) ≡ 0 (mod 8) for n ≥ 0obtained by Hirschhorn and Sellers,
and Fortin, Jacob and Mathieu, we confirm the conjecture of Hirschhorn and Sellers.
AMS Classification:
11P83, 05A17
Keywords:
overpartition, congruence, theta function, dissection formula, (p, k)-parametrization
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