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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Elementary proof of a formula of Ramanujan


Author: Robert L. Lamphere
Journal: Proc. Amer. Math. Soc. 91 (1984), 416-420
MSC: Primary 33A15; Secondary 05A30, 26A42
MathSciNet review: 744641
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Abstract: In this paper we use only elementary calculus to prove Ramanujan's integral formula

$\displaystyle \int_0^\infty {{x^{m - 1}}} \frac{{(1 + abx)(1 + a{b^2}x) \cdots ... ...nfty {\frac{{(1 - {b^{k - m}})(1 - a{b^k})}}{{(1 - {b^k})(1 - a{b^{k - m}})}}} $

where $ m$, $ a$ and $ b$ are positive with $ b < 1$ and $ a < {b^{m - 1}}$.

References [Enhancements On Off] (What's this?)

  • [1] Richard Askey, Ramanujan’s extensions of the gamma and beta functions, Amer. Math. Monthly 87 (1980), no. 5, 346–359. MR 567718 (82g:01030), http://dx.doi.org/10.2307/2321202
  • [2] Joseph Edwards, Treatise on integral calculus, vol. 2, reprinted Chelsea, New York, 1922.
  • [3] G. H. Hardy, Proof of a formula of Mr. Ramanujan, Messenger of Math. 44 (1915), 18-21.
  • [4] -, Pure mathematics, 1952; reprinted Cambridge University Press, London, 1963.
  • [5] S. Ramanujan, Collected papers, Chelsea, New York, 1962.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0744641-1
PII: S 0002-9939(1984)0744641-1
Article copyright: © Copyright 1984 American Mathematical Society