HYPERGEOMETRIC SERIES ACCELERATION VIA THE WZ METHOD Tewodros Amdeberhan and Doron Zeilberger Summary: HYPERGEOMETRIC SERIES ACCELERATION VIA THE WZ METHOD Tewodros Amdeberhan and Doron Zeilberger Department of Mathematics, Temple University, Philadelphia PA 19122, USA tewodros@math.temple.edu, zeilberg@math.temple.edu 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) = s\Gamma1 X i=0 Collections: Mathematics