### 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 |
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Title of host publication | SURVEYS IN NUMBER THEORY |

Editors | KRISHNASWAMI ALLADI |

Pages | 21-42 |

Number of pages | 22 |

DOIs | |

Publication status | Published - 2008 Dec 1 |

### Publication series

Name | Developments in Mathematics |
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Volume | 17 |

ISSN (Print) | 1389-2177 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## 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