Infinite families of congruences for 3-regular partitions with distinct odd parts (Q2660492)

Let \(pod_3(n)\) denote the number of 3-regular partitions with distinct odd parts of a nonnegative integer \(n\). In the paper under review, the author establishes some Ramanujan-type congruences modulo 2 and 3 for \(pod_3(n)\). Explicitly, the author proves that for a prime \(p\) larger than 4 with \((\frac{-2}{p})=-1\) (here \((\frac{\cdot}{\cdot})\) denotes the Legendre symbol), and for any nonnegative integer \(\alpha\), the following Ramanujan-type congruences modulo 2 for \(pod_3(n)\) hold: \[ \sum_{n=0}^{\infty}pod_3\left(2p^{2\alpha}n+\frac{p^{2\alpha}-1}{4}\right)q^n\equiv (q;q)_{\infty}(q^2;q^2)_{\infty}\ \pmod 2. \] For any nonnegative integer \(\alpha\), the author also obtains the following Ramanujan-type congruences modulo 3 for \(pod_3(n)\): \[ \sum_{n=0}^{\infty}pod_3\left(3^{2\alpha+2}n+\frac{3^{2\alpha+2}-1}{4}\right)(-1)^{3^{2\alpha+2}n}q^n\equiv \frac{(q^2;q^2)_{\infty}^4}{(q;q)_{\infty}^2}\ \pmod 3. \] The proofs are based on some q-identities. Standard generating function manipulations and binomial theorem are also used. As corollaries, infinite families of congruences modulo 2 and 3 for \(pod_3(n)\) are obtained.