On a formula proposed by S. Ramanujan
نویسندگان
چکیده
منابع مشابه
Analogues of a Transformation Formula of Ramanujan
We derive two new analogues of a transformation formula of Ramanujan involving the Gamma and Riemann zeta functions present in the Lost Notebook. Both involve infinite series consisting of Hurwitz zeta functions and yield modular relations. As a special case of the first formula, we obtain an identity involving polygamma functions given by A.P. Guinand and as a limiting case of the second formu...
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Let pr,s(n) denote the number of partitions of a positive integer n into parts containing no multiples of r or s, where r > 1 and s > 1 are square-free, relatively prime integers. We use classical methods to derive a Hardy-Ramanujan-Rademacher-type infinite series for pr,s(n).
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During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Based on numerical computations, Van Hamme recently conjectured p-adic analogues to such formulae. Using a combination of ordinary and Gaussian hypergeometric series, we prove one of these conjectures. Mathematics Subject Classification (2000). Primary: 33C20; Secondary: 11S80.
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It is shown that Ramanujan could have proved a special case of his conjecture that his tau function is multiplicative without any recourse to modularity results.
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We deduce new q-series identities by applying inverse relations to certain identities for basic hypergeometric series. The identities obtained themselves do not belong to the hierarchy of basic hypergeomet-ric series. We extend two of our identities, by analytic continuation, to bilateral summation formulae which contain Ramanujan's 1 ψ 1 summation and a very-well-poised 4 ψ 6 summation as spec...
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ژورنال
عنوان ژورنال: Journal of Computational and Applied Mathematics
سال: 1991
ISSN: 0377-0427
DOI: 10.1016/0377-0427(91)90105-s