Generalisations of some theorems of Ramanujan

Ramanathan, K. G. (1988) Generalisations of some theorems of Ramanujan Journal of Number Theory, 29 (2). 118-137 . ISSN 0022-314X

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Let φk,N(x) = f(-x2k, -xN-2k)/f(-xk, -xN-k) = Σn = 0uk(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 uk(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/Nφk,Nx1/N) where ϱ runs through all Nth roots of unity, in terms of f(a, b) and, following Andrews, to use Gordon's theorem.

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Deposited On:12 Apr 2011 11:23
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