Poincaré series having arbitrary positive real weight (Q1318864)

The existence of Poincaré series for the weight \(1 < r < 2\) and for any general discrete subgroup of \(\text{SL}_ 2 (\mathbb{R})\) has been proved in this note. Based on this result the author obtains the following identity. Let \(4\mid N\) and \(r = {1 \over 2} + \ell\) with a positive integer \(\ell\). Let \(P_ m(z) = \sum^ \infty_{n=1} \widehat P_ m (n)e(nz)\) be the Fourier expansion of Poincaré series in the space of cusp forms of half-integral weight \(r\) and on the group \(\Gamma_ 0(N)\). Then \[ \widehat P_ n(n) = 2 \left\{ 1 + 2 \pi i^{-r} \sum_{N | c,\;c > 0} c^{-1} J_{r-1} \left( {4 \pi n \over c} \right) S(n,n,c) \right\} \] where \(J_{r-1} (z)\) is the Bessel function of order \(r-1\) and \(S(n,n,c)\) is the generalized Kloosterman sum. This is a new result in the case of weight 3/2.