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Ramanujan's formula for the logarithmic derivative of the gamma function

Published online by Cambridge University Press:  24 October 2008

David Bradley
David Bradley
Affiliation:
Department of Mathematics & Statistics, Simon Fraser University, Burnaby, B.C. V5A 1S6Canada e-mail address: dbradley@cecm.sfu.ca
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Abstract

We prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in the notebooks [5]. The formula has a number of very interesting consequences which we derive, including an elegant hyperbolic summation, Ramanujan's formula for the Riemann zeta function evaluated at the odd positive integers, and new formulae for Euler's constant γ.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 3 , October 1996 , pp. 391 - 401
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[1]Abramowitz, M. and Stegun, I.. Handbook of mathematical functions (Dover, 1972).Google Scholar
[2]Berndt, Bruce C.. Modular transformations and generalizations of several formulae of Ramanujan, Rocky Mt. J.Math. 7 (1977), 147–189.CrossRefGoogle Scholar
[3]Berndt, Bruce C.. Ramanujan's Notebooks Part II (Springer-Verlag, 1989).CrossRefGoogle Scholar
[4]Berndt, Bruce C.. Ramanujan's Notebooks Part IV (Springer-Verlag, 1994).CrossRefGoogle Scholar
[5]Ramanujan, S.. Notebooks (2 Volumes) (Tata Institute of Fundamental Research, 1957).Google Scholar