  We develop a unified approach to the construction of the hyperbolic and
elliptic Eisenstein series on a finite volume hyperbolic Riemann surface.
Specifically, we derive expressions for the hyperbolic and elliptic Eisenstein
series as integral transforms of the kernel of a wave operator. Established
results in the literature relate the wave kernel to the heat kernel, which
admits explicit construction from various points of view. Therefore, we obtain
a sequence of integral transforms which begins with the heat kernel, obtains a
Poisson and wave kernel, and then yields the hyperbolic and elliptic Eisenstein
series. In the case of a non-compact finite volume hyperbolic Riemann surface,
we finally show how to express the parabolic Eisenstein series in terms of the
integral transform of a wave operator.
