## Experimental Mathematics

### Empirically determined Apéry-like formulae for {$\zeta(4n+3)$}

#### 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.]

#### Article information

Source
Experiment. Math., Volume 6, Issue 3 (1997), 181-194.

Dates
First available in Project Euclid: 17 March 2003

https://projecteuclid.org/euclid.em/1047920419

Mathematical Reviews number (MathSciNet)
MR1481588

Zentralblatt MATH identifier
0887.11037

Subjects
Primary: 11Y60: Evaluation of constants
Secondary: 11M06: $\zeta (s)$ and $L(s, \chi)$

#### Citation

Borwein, Jonathan; Bradley, David. Empirically determined Apéry-like formulae for {$\zeta(4n+3)$}. Experiment. Math. 6 (1997), no. 3, 181--194. https://projecteuclid.org/euclid.em/1047920419