  In this paper we give explicit closed forms for the semi-reproducing kernels
associated with thinplate spline interpolation on the sphere. Polyharmonic or
thinplate splines for ${\mathbb R}^d$ were introduced by Duchon and have become
a widely used tool in myriad applications. The analogues for ${\mathbb
S}^{d-1}$ are the thin plate splines for the sphere. The topic was first
discussed by Wahba in the early 1980's, for the ${\mathbb S}^2$ case. Wahba
presented the associated semi-reproducing kernels as infinite series. These
semi-reproducing kernels play a central role in expressions for the solution of
the associated spline interpolation and smoothing problems. The main aims of
the current paper are to give a recurrence for the semi-reproducing kernels,
and also to use the recurrence to obtain explicit closed form expressions for
many of these kernels. The closed form expressions will in many cases be
significantly faster to evaluate than the series expansions. This will enhance
the practicality of using these thinplate splines for the sphere in
computations.
