Congruences for modular forms and generalized Frobenius partitions

TL;DR

This paper explores congruences and parity properties of generalized Frobenius partitions using modular forms, extending classical results and establishing infinitely many congruences under certain conditions.

Contribution

It proves the existence of infinitely many congruences for generalized Frobenius partition functions modulo primes coprime with 6k, and generalizes results on modular form coefficients parity.

Findings

Infinitely many congruences for cϕ_k(n) modulo ℓ with gcd(ℓ,6k)=1

Results on the parity of cϕ_k(n)

Generalization of Ono's work on modular form coefficients

Abstract

The partition function is known to exhibit beautiful congruences that are often proved using the theory of modular forms. In this paper, we study the extent to which these congruence results apply to the generalized Frobenius partitions defined by Andrews. In particular, we prove that there are infinitely many congruences for $c\phi_k(n)$ modulo $\ell,$ where $\gcd(\ell,6k)=1,$ and we also prove results on the parity of $c\phi_k(n).$ Along the way, we prove results regarding the parity of coefficients of weakly holomorphic modular forms which generalize work of Ono.