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Koshliakovs Formula and Guinands Formula in Ramanujans Lost Notebook

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Surveys in Number Theory

Part of the book series: Developments in Mathematics ((DEVM,volume 17))

Summary

On two pages in his lost notebook, Ramanujan recorded several theorems involving the modified Bessel function Kν(z). These include Koshliakov’s formula and Guinand’s formula, both connected with the functional equation of nonanalytic Eisenstein series, and both discovered by these authors several years after Ramanujan’s death. Other formulas, including one by K. Soni and two particularly elegant new results, are also stated without proof by Ramanujan. In this paper, we prove all the formulas claimed by Ramanujan on these two pages and conclude with a survey of related results.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA

    Bruce C. Berndt

  2. Department of Mathematics, Yonsei University, 134 Shinchon-dong, Seodaemun-gu, Seoul, 120-749, Korea

    Yoonbok Lee & Jaebum Sohn

Authors
  1. Bruce C. Berndt
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  2. Yoonbok Lee
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  3. Jaebum Sohn
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Correspondence to Bruce C. Berndt .

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© 2008 Springer-Verlag New York

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Berndt, B.C., Lee, Y., Sohn, J. (2008). Koshliakovs Formula and Guinands Formula in Ramanujans Lost Notebook. In: Surveys in Number Theory. Developments in Mathematics, vol 17. Springer, New York, NY. https://doi.org/10.1007/978-0-387-78510-3_2

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