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 SL(2,ℤ).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa171-1-1,
author = {Takumi Noda},
title = {On the functional properties of Bessel zeta-functions},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {1-13},
zbl = {06487222},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa171-1-1}
}
Takumi Noda. On the functional properties of Bessel zeta-functions. Acta Arithmetica, Tome 168 (2015) pp. 1-13. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa171-1-1/