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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

  

Source: Amdeberhan, Tewodros - Department of Mathematics, Tulane University

 

Collections: Mathematics