Koshliakov's formula and Guinand's formula in Ramanujan's lost notebook

Bruce C. Berndt, Yoonbok Lee, Jaebum Sohn

Research output: Chapter in Book/Report/Conference proceedingChapter

12 Citations (Scopus)

Abstract

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.

Original languageEnglish
Title of host publicationSURVEYS IN NUMBER THEORY
EditorsKRISHNASWAMI ALLADI
Pages21-42
Number of pages22
DOIs
Publication statusPublished - 2008 Dec 1

Publication series

NameDevelopments in Mathematics
Volume17
ISSN (Print)1389-2177

Fingerprint

Ramanujan's Lost Notebook
Ramanujan
Modified Bessel Functions
Eisenstein Series
Functional equation
Theorem

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Berndt, B. C., Lee, Y., & Sohn, J. (2008). Koshliakov's formula and Guinand's formula in Ramanujan's lost notebook. In KRISHNASWAMI. ALLADI (Ed.), SURVEYS IN NUMBER THEORY (pp. 21-42). (Developments in Mathematics; Vol. 17). https://doi.org/10.1007/978-0-387-78510-3_2
Berndt, Bruce C. ; Lee, Yoonbok ; Sohn, Jaebum. / Koshliakov's formula and Guinand's formula in Ramanujan's lost notebook. SURVEYS IN NUMBER THEORY. editor / KRISHNASWAMI ALLADI. 2008. pp. 21-42 (Developments in Mathematics).
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Berndt, BC, Lee, Y & Sohn, J 2008, Koshliakov's formula and Guinand's formula in Ramanujan's lost notebook. in KRISHNASWAMI ALLADI (ed.), SURVEYS IN NUMBER THEORY. Developments in Mathematics, vol. 17, pp. 21-42. https://doi.org/10.1007/978-0-387-78510-3_2

Koshliakov's formula and Guinand's formula in Ramanujan's lost notebook. / Berndt, Bruce C.; Lee, Yoonbok; Sohn, Jaebum.

SURVEYS IN NUMBER THEORY. ed. / KRISHNASWAMI ALLADI. 2008. p. 21-42 (Developments in Mathematics; Vol. 17).

Research output: Chapter in Book/Report/Conference proceedingChapter

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Berndt BC, Lee Y, Sohn J. Koshliakov's formula and Guinand's formula in Ramanujan's lost notebook. In ALLADI KRISHNASWAMI, editor, SURVEYS IN NUMBER THEORY. 2008. p. 21-42. (Developments in Mathematics). https://doi.org/10.1007/978-0-387-78510-3_2