  Some rapidly convergent formulae for special values of the Riemann zeta
function are given. We obtain a generating function formula for zeta(4n+3)
which generalizes Apery's series for zeta(3), and appears to give the best
possible series relations of this type, at least for n<12. The formula reduces
to a finite but apparently non-trivial combinatorial identity. The identity is
equivalent to an interesting new integral evaluation for the central binomial
coefficient. We outline a new technique for transforming and summing certain
infinite series. We also derive a beautiful formula which provides strange
evaluations of a large new class of non-terminating hypergeometric series. Our
main results are shown to be equivalent. At the time this article was submitted
for publication back in 1996, these results were only conjectures, but they
have subsequently been proved as a result of work due to Gert Almkvist and
Andrew Granville.
