Proof of a conjecture on a congruence modulo 243 for overpartitions
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Proof of a conjecture on a congruence modulo 243 for overpartitions Xiaoqian Huang1 · Olivia X. M. Yao1
© Akadémiai Kiadó, Budapest, Hungary 2019
Abstract Let p(n) ¯ denote the number of overpartitions of n. Recently, numerous congruences modulo powers of 2, 3 and 5 were established regarding p(n). ¯ In particular, Xia discovered several infinite families of congruences modulo 9 and 27 for p(n). ¯ Moreover, Xia conjectured that for n ≥ 0, p(96n ¯ + 76) ≡ 0 (mod 243). In this paper, we confirm this conjecture by using theta function identities and the ( p, k)-parametrization of theta functions. Keywords Overpartitions · Congruences · Theta functions Mathematics Subject Classification 11P83 · 05A17
1 Introduction The aim of this paper is to prove a conjecture on a congruence modulo 243 for the number of overpartitions of n given by Xia [9]. Overpartitions are ordinary partitions extended by allowing a possible overline designation on the first (or equivalently, the final) occurrence of a part. For instance, there are 8 overpartitions of 3: ¯ 2 + 1, 2¯ + 1, 2 + 1, ¯ 2¯ + 1, ¯ 1 + 1 + 1, 1¯ + 1 + 1, 3, 3, As usual, let p(n) ¯ denote the number of overpartitions of n, and define p(0) ¯ = 1. For example, the above example shows p(3) ¯ = 8. Since its introduction in [5], the construction of overpartition has been very popular, and has led to a number of studies in q-series, partition theory, number theory, modular and mock modular forms. The generating function for p(n) ¯ was given by Corteel and Lovejoy [5]:
B
Olivia X. M. Yao [email protected] Xiaoqian Huang [email protected]
1
Department of Mathematics, Jiangsu University, Zhenjiang 212013, Jiangsu, People’s Republic of China
123
X. Huang, O. X. M. Yao ∞
n p(n)q ¯ =
n=0
f2 , f 12
where and throughout this paper, f k is defined by fk =
∞
(1 − q nk ) for any positive integer k.
n=1
Recently, a number of congruences modulo powers of 2, 3 and 5 for p(n) ¯ have been discovered, see for example, Hirschhorn and Sellers [7], Xia [9], Xia and Yao [10], Xia and Zhang [12] and Yao [13]. Very recently, Xia [9] established various infinite families of congruences modulo 9 and 27 for p(n). ¯ Furthermore, he presented the following conjecture: Conjecture 1.1 For n ≥ 0, p(96n ¯ + 76) ≡ 0 (mod 35 )
(1.1)
In this paper, we establish the generating function modulo 243 for p(n) ¯ and then employ the the ( p, k)-parametrization of theta functions due to Alaca, Alaca and Williams [1,2,8] to prove Conjecture 1.1.
2 Preliminaries In order to establish the generating function for p(96n ¯ + 76) modulo 243, we collect some preliminary results in this section. From Entry 25 (v) and (vi) on page 40 in Berndt’s book [4], f 12 =
f 2 f 85 2 f 42 f 16
2 f 2 f 16 , f8
− 2q
f5 f2 f2 1 = 5 8 2 + 2q 45 16 2 f1 f 2 f 16 f2 f8
(2.1) (2.2)
and f 14 =
f 410 f 22 f 84
− 4q
f 22 f 84 f 42
.
(2.3)
Replacing q by q 3 in (2.1) yields f 32 =
5 f 6 f 24 2 f2 f 12 48
− 2q 3
2 f 6 f 48 . f 24
Hirschhorn, Garvan and Borwein [6] derived the following 2-dissection formula fo
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