  Poincar\'e in 1911 and Petersson in 1932 gave the now classical expression
for the parabolic Fourier coefficients of holomorphic Poincar\'e series in
terms of Bessel functions and Kloosterman sums. Later, in 1941, Petersson
introduced hyperbolic and elliptic Fourier expansions of modular forms and the
associated hyperbolic and elliptic Poincar\'e series. In this paper we express
the hyperbolic Fourier coefficients of Poincar\'e series, of both parabolic and
hyperbolic type, in terms of hypergeometric series and Good's generalized
Kloosterman sums. In an explicit example for the modular group, we see that the
hyperbolic Kloosterman sum corresponds to a sum over lattice points on a
hyperbola contained in an ellipse. This allows for numerical computation of the
hyperbolic Fourier coefficients.
