Hypergeometric series acceleration via the WZ method

Based on the WZ method, some series acceleration formulas are given. These formulas allow us to write down an infinite family of parametrized identities from any given identity of WZ type. Further, this family, in the case of the Zeta function, gives rise to many accelerated expressions for ζ(3). AMS Subject Classification: Primary 05A We recall [Z] that a discrete function A(n,k) is called Hypergeometric (or Closed Form (CF)) in two variables when the ratios A(n + 1, k)/A(n, k) and A(n, k + 1)/A(n, k) are both rational functions. A discrete 1-form ω = F (n, k)δk + G(n, k)δn is a WZ 1-form if the pair (F,G) of CF functions satisfies F (n+ 1, k)− F (n, k) = G(n, k + 1)−G(n, k). We use: N and K for the forward shift operators on n and k, respectively. ∆n := N−1, ∆k := K−1. Consider the WZ 1-form ω = F (n, k)δk+G(n, k)δn. Then, we define the sequence ωs, s = 1, 2, 3, . . . of new WZ 1-forms: ωs := Fsδk +Gsδn; where Fs(n, k) = F (sn, k) and Gs(n, k) = s−1 ∑ i=0 G(sn+ i, k). Proposition: ωs is WZ, for all s. Proof: (a) ωs is closed: ∆nFs = F (s(n+ 1), k)− F (sn, k) = s−1 ∑ i=0 ( F (sn + i+ 1, k)− F (sn+ i, k) )