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On the functional properties of Bessel zeta-functions

Volume 171 / 2015

Takumi Noda Acta Arithmetica 171 (2015), 1-13 MSC: Primary 11M41; Secondary 11F11. DOI: 10.4064/aa171-1-1

Abstract

Zeta-functions associated with modified Bessel functions are introduced as ordinary Dirichlet series whose coefficients are $J$-Bessel and $K$-Bessel functions. Integral representations, transformation formulas, a power series expansion involving the Riemann zeta-function and a recurrence formula are given. The inverse Laplace transform of Weber's first exponential integral is the basic tool to derive the integral representations. As an application, we give a new proof of the Fourier series expansion of the Poincaré series attached to ${\rm SL}(2,\mathbb {Z})$.

Authors

  • Takumi NodaCollege of Engineering
    Nihon University
    1 Nakagawara, Tokusada, Tamuramachi
    Koriyama, Fukushima 963-8642, Japan
    e-mail

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