# Congruences in fractional partition functions

## TL;DR

This paper extends known congruences for fractional partition functions by leveraging eta function representations and Hecke algebra, achieving higher power moduli and new congruences.

## Contribution

It introduces methods to lift congruences to higher powers of primes and generates new congruences for specific cases using advanced modular form techniques.

## Key findings

- Extended congruences to higher prime powers.
- Generated new congruences for d=2 case.
- Utilized eta function representations and Hecke algebra.

## Abstract

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 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.