Quadratic forms and congruences for ℓ-regular partitions modulo 3, 5 and 7

Abstract

Let bℓ(n) be the number of ℓ-regular partitions of n. We show that the generating functions of bℓ(n) with ℓ=3,5,6,7 and 10 are congruent to the products of two items of Ramanujan's theta functions ψ(q), f(−q) and (q;q)∞3 modulo 3, 5 and 7. So we can express these generating functions as double summations in q. Based on the properties of binary quadratic forms, we obtain vanishing properties of the coefficients of these series. This leads to several infinite families of congruences for bℓ(n) modulo 3, 5 and 7.