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

Let bℓ(n)bℓ(n) be the number of ℓ-regular partitions of n  . We show that the generating functions of bℓ(n)bℓ(n) with ℓ=3,5,6,7ℓ=3,5,6,7 and 10 are congruent to the products of two items of Ramanujan's theta functions ψ(q)ψ(q), f(−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)bℓ(n) modulo 3, 5 and 7.