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.
|Title of host publication||SURVEYS IN NUMBER THEORY|
|Number of pages||22|
|Publication status||Published - 2008|
|Name||Developments in Mathematics|
Bibliographical noteFunding 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.
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