Generalisations of some theorems of Ramanujan

Abstract

Let φ_k,N(x) = f(-x^2k, -x^N-2k)/f(-x^k, -x^N-k) = Σ_{n = 0}^∞u_k(n)xⁿ, where f(a, b) = f(b, a) = 1 + ∑_{π = 1}^∞(ab)^{n(n -1)/2(aπ+ bπ)}. The main objective of the present paper is to show that for general N satisfying (N, 6) = 1, 1 ≤ k < N/2, (k, N) = 1 and 0 ≤ 1 < N, there exists an integer n₀ ≥ 0 depending on N, k, and l such that u_k(Nn + l), n ≥ n₀, all have the same sign. We determine explicity the value of n₀ and also the sign. The method, like Ramanujan's is to express (1/N)Σϱϱ^-lx^-l/Nφ_k,N(ϱ_x^1/N) where ϱ runs through all Nth roots of unity, in terms of f(a, b) and, following Andrews, to use Gordon's theorem.