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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Official URL: http://linkinghub.elsevier.com/retrieve/pii/002231...

Related URL: http://dx.doi.org/10.1016/0022-314X(88)90096-0

## Abstract

Let φ_{k,N}(x) = f(-x^{2k}, -x^{N-2k})/f(-x^{k}, -x^{N-k}) = Σ_{n = 0}^{∞}u_{k}(n)x^{n}, 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 n_{0} ≥ 0 depending on N, k, and l such that u_{k}(Nn + l), n ≥ n_{0}, all have the same sign. We determine explicity the value of n_{0} and also the sign. The method, like Ramanujan's is to express (1/N)Σϱϱ^{-l}x^{-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.

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ID Code: | 36113 |

Deposited On: | 12 Apr 2011 11:23 |

Last Modified: | 12 Apr 2011 11:23 |

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