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An Integral Analogue of Theta Functions and Gauss Sums in Ramanujan's Lost Notebook

Published online by Cambridge University Press:  01 September 2009

BRUCE C. BERNDT  and
PING XU
BRUCE C. BERNDT
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, IL 61801, U.S.A. e-mail: berndt@illinois.edu
PING XU
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, IL 61801, U.S.A. e-mail: pingxu2@illinois.edu
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Abstract

One page in Ramanujan's lost notebook is devoted to claims about a certain integral with two parameters. One claim gives an inversion formula for the integral that is similar to the transformation formula for theta functions. Other claims are remindful of Gauss sums. In this paper we prove all the claims made by Ramanujan about this integral.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 2 , September 2009 , pp. 257 - 265
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

REFERENCES

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