Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: học: HYPERGEOMETRIC SERIES ACCELERATION VIA THE WZ METHOD | HYPERGEOMETRIC SERIES ACCELERATION VIA THE WZ METHOD Tewodros Amdeberhan and Doron Zeilberger Department of Mathematics Temple University Philadelphia PA 19122 USA tewodros@ zeilberg@ Submitted Sept 5 1996. Accepted Sept 12 1996 Dedicated to Herb Wilf on his one million-first 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 C 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 1-form F n k ỗk G n k ỗn is a WZ 1-form if the pair F G of CF functions satisfies F n 1 k F n k G n k 1 G n k . We use N and K for the forward shift operators on n and k respectively. An N 1 Ak K 1. Consider the WZ 1-form F n k ỗk G n k ỗn. Then we define the sequence s s 1 2 3 . of new WZ 1-forms s Fsỗk Gsỗn where s 1 Fs n k F sn k and Gs n k X G sn i k . i 0 Proposition s is WZ for all s. Proof a s is closed AnFs F s n 1 k F sn k X F sn i 1 k F sn i k i 0 x G sn i k 1 G sn i k i 0 s 1 s 1 G sn i k 1 G sn i k i 0 Ak Gs. i 0 Typeset by AviS-TgX 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 4 2 1997 R3 2 Note that since is a WZ it has the form Z f n k P n k ỗk Q n k ỗù for some CF f and some polynomials P and Q. b s has the form Indeed s can be rewritten as s f sn k p sn k ỗk X f sn k- Q sn i k ỗn i 0 f sn f sn k P sn k ỗk R n k ỗn where R n k is a rational function and f sn k is still CF. Hence after pulling out a common denominator we see that s too has the form . This proves the Proposition. Theorem 1 Z Theorem 7 For any WZ pair F G oo 1 n 1 X G n 0 X F n n 1 G n 1 n 1 lim X F n k n 0 n 1 k 0 whenever both side converge.
