  This paper presents a parametric family of compactly-supported positive
semidefinite kernels aimed to model the covariance structure of second-order
stationary isotropic random fields defined in the $d$-dimensional Euclidean
space. Both the covariance and its spectral density have an analytic expression
involving the hypergeometric functions ${}_2F_1$ and ${}_1F_2$, respectively,
and four real-valued parameters related to the correlation range, smoothness
and shape of the covariance. The presented hypergeometric kernel family
contains, as special cases, the spherical, cubic, penta, Askey, generalized
Wendland and truncated power covariances and, as asymptotic cases, the
Mat\'ern, Laguerre, Tricomi, incomplete gamma and Gaussian covariances, among
others. The parameter space of the univariate hypergeometric kernel is
identified and its functional properties -- continuity, smoothness, transitive
upscaling (mont\'ee) and downscaling (descente) -- are examined. Several sets
of sufficient conditions are also derived to obtain valid stationary bivariate
and multivariate covariance kernels, characterized by four matrix-valued
parameters. Such kernels turn out to be versatile, insofar as the direct and
cross-covariances do not necessarily have the same shapes, correlation ranges
or behaviors at short scale, thus associated with vector random fields whose
components are cross-correlated but have different spatial structures.
