  We determine the current exchange amplitudes for free totally symmetric
tensor fields $\vf_{\mu_1 ... \mu_s}$ of mass $M$ in a $d$-dimensional $dS$
space, extending the results previously obtained for $s=2$ by other authors.
Our construction is based on an unconstrained formulation where both the
higher-spin gauge fields and the corresponding gauge parameters $\Lambda_{\mu_1
>... \mu_{s-1}}$ are not subject to Fronsdal's trace constraints, but
compensator fields $\alpha_{\mu_1 ... \mu_{s-3}}$ are introduced for $s > 2$.
The free massive $dS$ equations can be fully determined by a radial dimensional
reduction from a $(d+1)$-dimensional Minkowski space time, and lead for all
spins to relatively handy closed-form expressions for the exchange amplitudes,
where the external currents are conserved, both in $d$ and in $(d+1)$
dimensions, but are otherwise arbitrary. As for $s=2$, these amplitudes are
rational functions of $(ML)^2$, where $L$ is the $dS$ radius. In general they
are related to the hypergeometric functions $_3F_2(a,b,c;d,e;z)$, and their
poles identify a subset of the "partially-massless" discrete states, selected
by the condition that the gauge transformations of the corresponding fields
contain some non-derivative terms. Corresponding results for $AdS$ spaces can
be obtained from these by a formal analytic continuation, while the massless
limit is smooth, with no van Dam-Veltman-Zakharov discontinuity.
