Congruences in fractional partition functions

Abstract

The coefficients of the generating function $(q;q{)}_{\infty }^{\alpha }$ 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 $p_{\frac{a}{b}}(\ell n+c)\equiv 0(\mathrm{mod}\ell )$ where $\ell$ is a prime such that $\ell \mid a-db$ for $d\in \{4,6,8,10,14,26\}$. Expanding upon their work, we use the representation of powers of the Dedekind-eta functions in linear sums of Hecke eigenforms and their lacunarity to raise the power of the modulus to higher powers of $\ell$. In addition, we generate congruences for when $d=2$ employing Hecke algebra.