Congruences modulo 7 for broken 3-diamond partitions
The notion of broken \(k\)-diamond partitions was introduced by \textit{G.E.~Andrews} and \textit{P.~Paule} in [Acta Arith. 126, No. 3, 281--294 (2007; Zbl 1110.05010)]. Let \(\Delta_k(n)\) denote the number of broken \(k\)-diamond partitions of \(n\) for a fixed positive integer \(k\). \textit{P. Paule} and \textit{S. Radu} [Ramanujan J. 23, No. 1-3, 409--416 (2010; Zbl 1218.05024)] conjectured that for any \(n\geq0\), \[ \Delta_3(343n+s)\equiv0\pmod{7},\quad\text{where}~s\in\{82,278,327\}. \] These congruences were confirmed by \textit{M.~Jameson} [Ann. Comb. 17, No. 2, 333--338 (2013; Zbl 1270.05009)] via the theory of modular forms. Moreover, she also proved that for any \(n\geq0\), \[ \Delta_3(343n+229)\equiv0\pmod{7}. \] By establishing a recurrence relation for the coefficients of certain infinite \(q\)-series product, \textit{E.X.W. Xia} [Ramanujan J. 40, No. 2, 389--403 (2016; Zbl 1415.11147)] gave a new proof of the aforementioned four congruences modulo 7 satisfied by \(\Delta_3(n)\). He also proved several infinite families of Ramanujan-type congruences modulo 7 for \(\Delta_3(n)\). For instance, for any \(n\), \(k\geq0\), \[ \Delta_3{\left(7\times4^{7k+4}n+\dfrac{77\times4^{7k+3}+1}{3}\right)}\equiv0\pmod{7}. \] Soon after, \textit{O.X.M. Yao} and \textit{Y. Wang} [Ramanujan J. 43, No. 3, 619--631 (2017; Zbl 1432.11151)] proved many infinite families of congruences modulo 7 for \(\Delta_3(n)\) by utilizing an identity due to Newman and \((p,k)\)-parametrization of theta functions. In particular, they proved that for any \(n\geq0\) and \(k\geq1\), \[ \Delta_3{\left(98\times4^kn+\dfrac{182\times4^{k-1}+1}{3}\right)}\equiv0\pmod{7}.\tag{1} \] In this paper under review, the authors not only provide a new proof of (1), but prove three similar congruences for \(\Delta_3(n)\). More specifically, they prove that for any \(n\geq0\) and \(k\geq1\), \[ \Delta_3{\left(98\times4^kn+\dfrac{t\times4^{k-1}+1}{3}\right)}\equiv0\pmod{7},\quad\text{where} ~t\in\{182,686,854,1022\}. \] The main ingredients of proofs are several identities involving theta functions \(\varphi(q)\) and \(\psi(q)\), two product-to-sum identities due to Ramanujan and Gordon, and a recurrence relation for the coefficients of \(\displaystyle{\prod\limits_{n=1}^\infty}(1-q^n)^4(1-q^{2n})^6\).
scientific article; zbMATH DE number 7140963