Empirically Determined Apéry-Like Formulae for ζ(4n+3)
- 1 January 1997
- journal article
- research article
- Published by Taylor & Francis in Experimental Mathematics
- Vol. 6 (3) , 181-194
- https://doi.org/10.1080/10586458.1997.10504608
Abstract
Somerapidly convergent formulae for special values of the Riemann zeta function are given. We obtain a generating function formula for ζ(4n+3) that generalizes Apervs seriesfor ζ(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 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. Seenote on page 194.]Keywords
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