Summary: HYPERGEOMETRIC SERIES ACCELERATION VIA THE WZ METHOD
Tewodros Amdeberhan and Doron Zeilberger
Department of Mathematics, Temple University, Philadelphia PA 19122, USA
Submitted: Sept 5, 1996. Accepted: Sept 12, 1996
Dedicated to Herb Wilf on his one millionfirst birthday
Abstract. 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 i(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 1form ! = F (n; k)ffik + G(n; k)ffin is a WZ 1form if the pair (F,G) of CF functions satisfies
F (n + 1; k) \Gamma F (n; k) = G(n; k + 1) \Gamma G(n; k).
We use: N and K for the forward shift operators on n and k, respectively. \Delta n := N \Gamma 1, \Delta k := K \Gamma 1.
Consider the WZ 1form ! = F (n; k)ffik +G(n; k)ffin. Then, we define the sequence ! s ; s = 1; 2; 3; : : :
of new WZ 1forms: ! s := F s ffi k +G s ffi n; where
F s (n; k) = F (sn; k) and G s (n; k) =