On the functional properties of Bessel zeta-functions

Takumi Noda¹

¹ College of Engineering Nihon University 1 Nakagawara, Tokusada, Tamuramachi Koriyama, Fukushima 963-8642, Japan

Acta Arithmetica, Tome 171 (2015) no. 1, pp. 1-13

Résumé

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 ${\rm SL}(2,\mathbb {Z})$.

Détail

DOI : 10.4064/aa171-1-1

Keywords: zeta functions associated modified bessel functions introduced ordinary dirichlet series whose coefficients j bessel k bessel functions integral representations transformation formulas power series expansion involving riemann zeta function recurrence formula given inverse laplace transform webers first exponential integral basic tool derive integral representations application proof fourier series expansion poincar series attached mathbb