Some rapidly convergent formulae for special
values of the Riemann zeta function are given.
We obtain a generating function formula for \zet$(4n+3)$
that generalizes Apéry's series for \zet$(3)$, and appears
to give the best possible series relations of this type,
at least for n{\mathversion{normal}$\,<\,$}12.
The formula reduces to a finite but
apparently nontrivial 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 formula that
provides strange evaluations of a large new class of nonterminating
hypergeometric series.
¶ [Editor's Note: The beautiful formulas in this paper are no longer
conjectural. See note on page 194.]