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

Bruce C. Berndt; Yoonbok Lee; Jaebum Sohn

doi:10.1007/978-0-387-78510-3_2

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

Bruce C. Berndt, Yoonbok Lee, Jaebum Sohn

Department of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter

19 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 language	English
Title of host publication	SURVEYS IN NUMBER THEORY
Editors	KRISHNASWAMI ALLADI
Pages	21-42
Number of pages	22
DOIs	https://doi.org/10.1007/978-0-387-78510-3_2
Publication status	Published - 2008

Publication series

Name	Developments in Mathematics
Volume	17
ISSN (Print)	1389-2177

Bibliographical note

Funding Information:
This research was supported by the Marsden Fund, administered by the Royal Society of New Zealand, and the University of Otago. Discussions with P.O. Koons, R. McDowall, and R.J. Norris helped to develop ideas, and i.M. Turnbull and P.J. Forsyth were helpful in sharing ideas on landscape evolution. Constructive reviews by R. McDowall and a.S. Jayko helped to improve the presentation and substance of the manuscript.

All Science Journal Classification (ASJC) codes

Mathematics(all)

Access to Document

10.1007/978-0-387-78510-3_2

Cite this

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

Abstract

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