Congruences for modular forms and generalized Frobenius partitions

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 (Mem Am Math Soc 49(301):iv+44, 1984). 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 (J Reine Angew Math 472:1–15, 1996).