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

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.

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.

A Class of Conjectured Series Representations for 1/π

Using the second conjecture in the paper [10] of the author and inspired by the theory of modular functions we find a method which allows us to obtain explicit formulae, involving η or θ functions, for the parameters of a class of series for 1/π. As in [10], the series considered in this paper include Ramanujan’s series as well as those associated with the Domb numbers and Apéry numbers. 1 A sp...

Card Deals, Lattice Paths, Abelian Words and Combinatorial Identities

We give combinatorial interpretations of several related identities associated with the names Barrucand, Strehl and Franel, including one for the Apéry numbers, ∑n k=0 (n k )(n+k k ) ∑k j=0 (k j 3 = ∑n k=0 (n k )2(n+k k 2 . The combinatorial constructs employed are derangement-type card deals as introduced in a previous paper on Barrucand’s identity, labeled lattice paths and, following a comme...

Simultaneous Generation of Koecher and Almkvist-Granville's Apéry-Like Formulae