Empirically Determined Apéry-Like Formulae for ζ(4n+3)

M ay 2 00 5 Empirically Determined Apéry - Like Formulae for ζ ( 4 n + 3 )

Abstract. Some rapidly convergent formulae for special values of the Riemann Zeta function are given. We obtain a generating function formula for ζ(4n+3) which generalizes Apéry’s series for 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 non-trivial combinatorial identity. The identity is equivalent ...

6 Experimental Determination of Apéry - Like Identities for ζ ( 2 n + 2 )

We document the discovery of two generating functions for ζ(2n + 2), analogous to earlier work for ζ(2n + 1) and ζ(4n + 3), initiated by Koecher and pursued further by Borwein, Bradley and others.

Experimental Determination of Apéry-like Identities for ς(2n + 2)

We document the discovery of two generating functions for ζ(2n + 2), analogous to earlier work for ζ(2n + 1) and ζ(4n + 3), initiated by Koecher and pursued further by Borwein, Bradley and others.

Simultaneous Generation for Zeta Values by the Markov-WZ Method

By application of the Markov-WZ method, we prove a more general form of a bivariate generating function identity containing, as particular cases, Koecher’s and Almkvist-Granville’s Apéry-like formulae for odd zeta values. As a consequence, we get a new identity producing Apéry-like series for all ζ(2n+ 4m+ 3), n,m ≥ 0, convergent at the geometric rate with ratio 2−10.

Borwein and Bradley's Apérv-Like Formulae for ζ(4n + 3)