  The spherical Fourier transform on a harmonic Hadamard manifold $(X^n, g)$ of
positive volume entropy is studied. If $(X^n, g)$ is of hypergeometric type,
namely spherical functions of $X$ are represented by the Gauss hypergeometric
functions, the inversion formula, the convolution rule together with the
Plancherel theorem are shown by the representation of the spherical functions
in terms of the Gauss hypergeometric functions. A geometric characterization of
hypergeometric type is derived in terms of volume density of geodesic spheres.
Geometric properties of $(X^n, g)$ are also discussed.
