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1997 Empirically determined Apéry-like formulae for {$\zeta(4n+3)$}
Jonathan Borwein, David Bradley
Experiment. Math. 6(3): 181-194 (1997).

Abstract

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.]

Citation

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Jonathan Borwein. David Bradley. "Empirically determined Apéry-like formulae for {$\zeta(4n+3)$}." Experiment. Math. 6 (3) 181 - 194, 1997.

Information

Published: 1997
First available in Project Euclid: 17 March 2003

zbMATH: 0887.11037
MathSciNet: MR1481588

Subjects:
Primary: 11Y60
Secondary: 11M06

Rights: Copyright © 1997 A K Peters, Ltd.

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Vol.6 • No. 3 • 1997
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