Two zeta functions contained in the Poincare series

Abstract

Two zeta-functions associated with the classical Poincare series attached to modular group are introduced. Integral representations, transformation formulas and some functional properties are given. As an application, we obtain two new proofs of the Fourier series expansion of the Poincare series attached to SL(2, ℤ). This manuscript is a summarized version of [T. Noda, On the functional properties of Bessel zeta-junctions, Acta Arith. 171, No.1(2015), 1-13.] and the forthcoming paper [T. Noda, The exponential type generating junction of the Riemann zeta-junction revisited, preprint.].