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Ramanujan's Continued Fraction

Published online by Cambridge University Press:  24 October 2008

G. N. Watson
G. N. Watson
Affiliation:
Trinity College
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Extract

The best of the theorems on continued fractions, to be found in Ramanujan's manuscript note-book may be stated as follows:

where there are eight gamma-functions in each product and the ambiguous signs are so chosen that the argument of each gamma-function contains one of the specified numbers of minus signs. Then

provided that one of the numbers β, γ δ, ε is equal to ± n, where n is a positive integer; and the products and sums on the right range over the numbers α β, γ δ, ε.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 31 , Issue 1 , January 1935 , pp. 7 - 17
Copyright
Copyright © Cambridge Philosophical Society 1935

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References

* Through an error in copying which occurred when I previously published an enunciation of the theorem, Journal London Math. Soc. 6 (1931), 146Google Scholar, the numbers in b 2 were inadvertently given as 9 and 10 instead of 13 and 14.

† Perron, O.. Die Lehre von den Kettenbrüchen (2. Aufl. 1929)Google Scholar.

* Hardy, G. H., “A chapter from Ramanujan's note-book”, Proc. Camb. Phil. Soc. 21 (1923), 492–503.Google Scholar

† Laguerre, E., “Sur la function ”, Bull. Soc. Math. France, 8 (1879), 36–52.Google Scholar

* Here and elsewhere it is supposed that the “numerators” and “denominators” of the convergents of a continued fraction are calculated without cancelling common factors, if such exist. Thus the numerators of the convergents of the continued fraction

are

* The properties of the original expressions show that their difference vanishes when α = 0, and so the difference contains α as a factor; similarly for β, γ, δ. Hence C n (α, β, γ, δ) is an integral function.