Empirically Determined Apéry-like Formulae for Zeta(4n+3)

Borwein, Jonathan M. and Bradley, David M. (1997) Empirically Determined Apéry-like Formulae for Zeta(4n+3). Experiment. Math., 6 (3). pp. 181-194.

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      Some rapidly convergent formulae for special values of the Riemann Zeta function are given. We obtain a generating function formula for $\z(4n+3)$ which generalizes Ap\'ery's series for $\z(3)$, 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. It should be emphasized that our main results are shown equivalent but are still only conjectures.

      Item Type: Article
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      Date Deposited: 24 Nov 2003
      Last Modified: 14 Sep 2014 21:57

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