Koshliakovs Formula and Guinands Formula in Ramanujans Lost Notebook

DOI10.1007/978-0-387-78510-3_2zbMath1183.33007OpenAlexW126860937MaRDI QIDQ3401682

Bruce C. Berndt, Jaebum Sohn, Yoonbok Lee

Publication date: 1 February 2010

Published in: Surveys in Number Theory (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/978-0-387-78510-3_2

zbMATH Keywords

modified Bessel functionsMaass wave formsdivisor functionsGuinand's formulaKoshliakov's formulanonanalytic Eisenstein series

Mathematics Subject Classification ID

Dedekind eta function, Dedekind sums (11F20) Bessel and Airy functions, cylinder functions, ({}_0F_1) (33C10) Automorphic forms, one variable (11F12)

Related Items (17)

Explicit transformations of certain Lambert series ⋮ A generalized modified Bessel function and a higher level analogue of the theta transformation formula ⋮ Two general series identities involving modified Bessel functions and a class of arithmetical functions ⋮ On the Epstein zeta function and the zeros of a class of Dirichlet series ⋮ Koshliakov kernel and identities involving the Riemann zeta function ⋮ Arithmetic and analysis of the series \(\sum_{n=1}^{\infty} \frac{1}{n} \sin \frac{x}{n} \). II ⋮ Character analogues of Cohen-type identities and related Voronoï summation formulas ⋮ Certain extensions of results of Siegel, Wilton and Hardy ⋮ Identities associated to a generalized divisor function and modified Bessel function ⋮ Transformation formulas associated with integrals involving the Riemann \(\Xi\)-function ⋮ Self-reciprocal functions, powers of the Riemann zeta function and modular-type transformations ⋮ Character analogues of theorems of Ramanujan, Koshliakov and Guinand ⋮ Series transformations and integrals involving the Riemann \(\Xi \)-function ⋮ Sums of squares and products of Bessel functions ⋮ The generalized modified Bessel function and its connection with Voigt line profile and Humbert functions ⋮ A general class of Voronoi's and Koshliakov–Ramanujan's summation formulas involvingd_k(n) ⋮ Extensions of Watson’s theorem and the Ramanujan–Guinand formula

Cites Work

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• On Bessel series expressions for some lattice sums: II
• SOME SUMMATIONS OVER THE LATTICE POINTS OF A CIRCLE (I)
• On Epstein's zeta function
• Ramanujan's Lost Notebook
• On Epstein's Zeta-function.
• Some Relations Associated with an Extension of Koshliakov's Formula
• Identities Involving the Coefficients of a Class of Dirichlet Series. III
• Identities Involving the Coefficients of a Class of Dirichlet Series. V
• Some Solutions of the Equation F (t ) = F (t ⁻¹ )
• ON THE SUMMATION FORMULAE OF VORONOÏ AND POISSON
• On Epstein's Zeta Function (I)
• SOME RAPIDLY CONVERGENT SERIES FOR THE RIEMANN ξ-FUNCTION

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