Proofs of some conjectures on the reciprocals of the Ramanujan-Gordon identities
Ramanujan and Gordon independently discovered the following two nice product-to-sum identities: \[ \sum_{n=-\infty}^{\infty}(6n+1)q^{n(3n+1)/2}=\frac{(q;q)_{\infty}^5}{(q^2;q^2)_{\infty}^2}, \] \[ \sum_{n=-\infty}^{\infty}(3n+1)q^{n(3n+2)}=\frac{(q;q)_{\infty}^2(q^4;q^4)_{\infty}^2}{(q^2;q^2)_{\infty}}, \] which are called the Ramanujan-Gordon identities. Define two partition functions \(RG_1(n)\) and \(RG_2(n)\) by \[ \sum_{n=0}^{\infty}RG_1(n)q^n=\frac{(q^2;q^2)_{\infty}^2}{(q;q)_{\infty}^5}, \] \[ \sum_{n=0}^{\infty}RG_2(n)q^n=\frac{(q^2;q^2)_{\infty}}{(q;q)_{\infty}^2(q^4;q^4)_{\infty}^2}, \] whose generating functions are the reciprocals of \(\sum_{n=-\infty}^{\infty}(6n+1)q^{n(3n+1)/2}\) and \(\sum_{n=-\infty}^{\infty}(3n+1)q^{n(3n+2)}\) respectively. In the paper under review, via Ramanujan's modular equation of fifth degree, the authors prove the following four congruences modulo 25 for \(RG_1(n)\) and \(RG_2(n)\): \begin{align*} RG_1(125n+74)&\equiv 0\pmod{25}, \\ RG_1(125n+124)&\equiv 0\pmod{25}, \\ RG_2(125n+92)&\equiv 0\pmod{25},\\ RG_2(125n+117)&\equiv 0\pmod{25}, \end{align*} which were conjectured by \textit{B. L. Lin} and \textit{A. Y. Wang} [Colloq. Math. 154, No. 1, 137--148 (2018; Zbl 1429.11190)]. In addition, the authors also establish some new congruences modulo 25 for \(RG_1(n)\) and \(RG_2(n)\). Their proof is based on Newman's identities. For example, they deduce that for any integer \(n\geq 0\), \[ RG_1\left(\frac{23375n(3n+1)}{2}+974\right)\equiv RG_1\left(\frac{23375n(3n+5)}{2}+24349\right)\equiv 0\pmod{25}. \] \[ RG_2(2000n(3n+1)+167)\equiv RG_2(2000n(3n+5)+4167)\equiv 0\pmod{25}. \]
scientific article; zbMATH DE number 7369443
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