The coefficients of the generating function \((q;q)^\alpha_\infty\) produce \(p_\alpha(n)\) for \(\alpha \in \mathbb{Q}\). In particular, when \(\alpha = -1\), the partition function is obtained. Recently, Chan and Wang identified and proved congruences of the form \(p_{\frac{a}{b}}(\ell n + c)\equiv 0 \pmod{\ell}\) where \(\ell\) is a prime such t...

Congruences in fractional partition functions