 
Summary: FASTER AND FASTER CONVERGENT SERIES FOR i(3)
Tewodros Amdeberhan
Department of Mathematics, Temple University, Philadelphia PA 19122, USA
tewodros@euclid.math.temple.edu
Submitted: April 8, 1996. Accepted: April 15, 1996
Abstract. Using WZ pairs we present accelerated series for computing i(3)
AMS Subject Classification: Primary 05A
Alf van der Poorten [P] gave a delightful account of Ap'ery's proof [A] of the irrationality of i(3). Using
WZ forms, that came from [WZ1], Doron Zeilberger [Z] embedded it in a conceptual framework.
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 pair
(F,G) of CF functions is a WZ pair if F (n + 1; k) \Gamma F (n; k) = G(n; k + 1) \Gamma G(n; k). In this paper, after
choosing a particular F (where its companion G is then produced by the amazing Maple package EKHAD
accompanying [PWZ]), we will give a list of accelerated series calculating i(3). Our choice of F is
F (n; k) = (\Gamma1) k k! 2 (sn \Gamma k \Gamma 1)!
(sn + k + 1)!(k + 1)
where s may take the values s=1,2,3, : : : [AZ] (the section pertaining to this can be found in
http://www.math.temple.edu/~tewodros). In order to arrive at the desired series we apply the following
result:
Theorem: ([Z], Theorem 7, p.596) For any WZ pair (F,G)
