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 million- rst birthday Abstract. Based on the WZ method, some series accelerationformulas are given. These formulas allow us to write down an in nite family of parametrizedidentities from any given identity of WZ type. Further, this family, in the case of the Zeta function, gives rise to many accelerated expressions for 3. AMS Subject Classi cation: Primary 05A We recall Z that a discrete function An,k is called Hypergeometric or Closed Form CF in two variables when the ratios An + 1;k=An;k and An;k + 1=An;k are both rational functions. A discrete 1-form ! = Fn;k k + Gn;k n is a WZ 1-form if the pair F,G of CF functions satis es Fn+1;k,Fn;k = Gn;k+1 ,Gn;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 ! = Fn;k k+Gn;k n. Then, we de ne the sequence !s;s = 1;2;3;::: of new WZ 1-forms: !s := Fs k + Gs n; where Fsn;k = Fsn;k and Gsn;k = s,1 Collections: Mathematics