H. S. Wilf and D. Zeilberger, "Rational functions certify combinatorial identities,"J. Amer. Math. Soc. 3 (1990), 147--158.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(4) where ordinary factorials (or the Gamma function) are applied only to constants on the right hand side. The advantage of rising factorials over ordinary ones is that the former do not rely on the Gamma function and are well defined in any field of characteristic zero. Wilf and Zeilberger [18] associate with (n ff) its shadow ( Gamman Gamma ff Gamma 1) which satisfies the same first order recurrence w.r.t. n. When ff = 2 Z the shadow is just a constant factor multiple of (n ff) the constant being Gamma(sin ff) while for ff 2 Z the shadow is complementary to (n ff) ....

H. S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc. 3 (1990) 147--158.

....z 2 =2n 2 = X n1 1 n 2 Gamma 2n n Delta Gamma 1 2 2 (1 Gamma z 2 =n 2 ) n Gamma1 Y m=1 1 Gamma z 2 m 2 : These have a slightly different flavour to the generalization given in Conjecture 1. Using the revolutionary method of Wilf, Zeilberger (and Ekhad) [9], Amdeberhan and Zeilberger [1] gave the following striking, and fast converging formula, i(3) 1 2 X n1 ( Gamma1) n Gamma1 (205n 2 Gamma 160n 32) n 5 Gamma 2n n Delta 5 ; amongst several others. Ackowledgements. Thanks are due to David Bradley for pointing us to several of the ....

H.S. Wilf, D. Zeilberger, Rational functions certify combinatorial identities, Jour. Amer.

....that if b n b n 1 is a rational function of n, then so is a n a n 1 = b n b n 1 1 1 b n 2 b n 1 . 3.4) Therefore Gosper s algorithm should be applied only when a n a n 1 is rational. The other recent development is the Wilf Zeilberger method for proving combinatorial identities [379, 380]. Given a conjectured identity, it provides an algorithmic procedure for verifying it. This method succeeds in a surprisingly wide range of cases. Typically, to prove an identity of the form # k U(n, k) S(n) n # 0 , 3.5) 10 where S(n) #= 0, Wilf and Zeilberger define F (n, k) U(n, ....

H. S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc., 3 (1990), pp. 147--158.

....=n 2 ) n Gamma1 Y m=1 1 Gamma z 2 m 2 : These formulas have a slightly different flavour from the generalization given in Conjecture 1. 7 June 1999 at 21:02 Almkvist and Granville: Borwein and Bradley s Ap ery like formulae for i(4n 3) 203 Using the revolutionary method of Wilf and Zeilberger [1990], Amdeberhan and Zeilberger [1997] found the striking, and fast converging, formula i(3) 1 2 X n1 ( Gamma1) n Gamma1 (205n 2 Gamma 160n 32) n 5 Gamma 2n n Delta 5 ; amongst several others. ACKNOWLEDGEMENTS Thanks are due to David Bradley for pointing us to several of the ....

H. S. Wilf and D. Zeilberger, "Rational functions certify combinatorial identities", J. Amer. Math. Soc. 3:1 (1990), 147--158.

....of each method. Submitted: December 2, 1994; Accepted: November 13, 1995 1. Introduction. The first two papers in this series raise the following obvious question: Why should anyone want to resurrect a method last used in 1797 to sum hypergeometric series when it is well known that the WZ method [13] has swept all before it. This latter state of affairs has been spelled out in delightful albeit idiosyncratic detail by Zeilberger in [16] and [17] The reader is urged to consult these references for the complete understanding of his philosophy. Perhaps the case can be put succinctly by ....

....the WZ method proves (2.4) by showing that (2.5) Gammam b Gamma c) Gammam a Gamma c)F 1 (m; r) Gamma (c m) Gammam a b Gamma c)F 1 (m 1; r) G 1 (m; r) Gamma G 1 (m; r Gamma 1) where (2. 6) G 1 (m; r) a r) b r)F 1 (m; r) is the auxiliary function called the certificate [13]. Equation (2.4) is then deduced by summing (2.5) from r = 0 to r = m 1. Thus I would suggest that neither method is innately superior in proving (2.1) Kummer s theorem [6; x2.3] cf. 2; x3] was the next example considered in our exposition of the Pfaff method. Again in this case, Pfaff s ....

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H. S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc., 3 (1990), 147--158.

....of each method. Submitted: December 2, 1994; Accepted: November 13, 1995 1. Introduction. The first two papers in this series raise the following obvious question: Why should anyone want to resurrect a method last used in 1797 to sum hypergeometric series when it is well known that the WZ method [13] has swept all before it. This latter state of a#airs has been spelled out in delightful albeit idiosyncratic detail by Zeilberger in [16] and [17] The reader is urged to consult these references for the complete understanding of his philosophy. Perhaps the case can be put succinctly by referring ....

....di#erencing of two sums while the WZ method proves (2.4) by showing that (2.5) m b c) m a c)F 1 (m, r) c m) m a b c)F 1 (m 1,r) G 1 (m, r) G 1 (m, r 1) where (2. 6) G 1 (m, r) a r) b r)F 1 (m, r) is the auxiliary function called the certificate [13]. Equation (2.4) is then deduced by summing (2.5) from r =0tor = m 1. Thus I would suggest that neither method is innately superior in proving (2.1) Kummer s theorem [6; 2.3] cf. 2; 3] was the next example considered in our exposition of the Pfa# method. Again in this case, Pfa# s method ....

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H. S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities,J.Amer. Math. Soc., 3 (1990), 147--158.

....one gets by applying qZeil the minimal recurrence of order 1. In[15] qZeil[ 1)k (1 qk) 2 q(k(3k 1) 2) qBinomial[2n,n k,q] k, Infinity,Infinity, n, 1] n 1 2 n Out[15] SUM[n] 1 q ) 1 q ) SUM[ 1 n] Paule s method is of special importance with respect to the theory of q WZ pairs [16]. There are various applications, 10] or [14] where summing the even part enables one to manufacture the dual or companion identities. We only mention two examples for which the q Zeilberger algorithm delivers a recurrence of increased order 3, namely the Rogers identity (31) and the ....

H. S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc. 3 (1990) 147--158.

.... qfac[q,q,3n 1] k, n 1, n, n, 1] Out[6] SUM[n] 1 60 3.3 The Companion Identity We already saw that qWZ pairs play an important role in q certification. Moreover, we can use qWZ pairs to get new identities for free , i.e. without too much additional effort (cf. Wilf and Zeilberger [21]) One of them is called the companion identity and is based on the symmetry of F and G in the qWZ equation (3.1.1) Theorem 3.3.1 (companion identity) Let F and G form a qWZ pair satisfying the following conditions: F) For each integer k, the limit f k : lim n 1 F (n; k) exists and is ....

.... Gamma G(n; k Gamma 1) F (n; k Gamma 1) F (n; k Gamma 1) F (n; k) By our assumptions we may replace F and G by e F and e G, respectively. Multiplying through by e F (n; k) proves that ( e F ; e G) form a qWZ pair. 64 As in the q = 1 case (cf. Gessel [7] Wilf [20] or Wilf and Zeilberger [21], 22] we introduce the operation of shadowing. Let, for instance, a(n) q; q) n for n 0. Then the defining property of a(n) is that it satisfies the recurrence equation a(n) 1 Gamma q n ) a(n Gamma 1) together with the initial condition a(0) 1. But why should we restrict ourselves ....

H.S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc., 3 (1990), 147--158.

....gets by applying qZeil the minimal recurrence of order 1. 17 In[15] qZeil[ 1)k (1 qk) 2 q(k(3k 1) 2) qBinomial[2n,n k,q] k, Infinity,Infinity, n, 1] n 1 2 n Out[15] SUM[n] 1 q ) 1 q ) SUM[ 1 n] Paule s method is of special importance with respect to the theory of q WZ pairs [16]. There are various applications, 10] or [14] where summing the even part enables one to manufacture the dual or companion identities. We only mention two examples for which the q Zeilberger algorithm delivers a recurrence of increased order 3, namely the Rogers identity (31) and the ....

H. S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc. 3 (1990) 147--158.

....1)V 2 2m 2n 2 V 4m 1 m 1 # j=0 ( 1 2 ) 2 j ( 5 4 ) j (1 m n) j (m n) j (1) 2 j ( 1 4 ) j ( 1 2 m n) j ( 3 2 m n) j . To finish the solution of Propp s Problem 1, we need only evaluate the sum in Theorem 15 in the case m = n. TodothisweusetheWilf Zeilberger(WZ)method [27]. the electronic journal of combinatorics 6 (1999) #R16 19 Lemma 16. n 1 # i=0 ( 1 2 ) 2 i ( 5 4 ) i (1 2n) i (2n) i (1) 2 i ( 1 4 ) i ( 1 2 2n) i ( 3 2 2n) i = 4n 1 3 . Proof. Let Q(n, i) 1 4n 1 ( 1 2 ) 2 i ( 5 4 ) i (1 2n) i (2n) i (1) 2 i ....

H. S. Wilf and D. Zeilberger, "Rational functions certify combinatorial identities,"J. Amer. Math. Soc. 3 (1990), 147--158.

.... expressions, which include products of rational functions, factorials, binomial coefficients, and exponentials, the summation problem has first been solved by Gosper [13] Other contributions are due to Abramov [1] Moenck [18] Karr [15, 16] Zeilberger [29, 28, 30] Wilf Zeilberger [26, 27], Lisonek et al. 17] Pirastu Strehl [22] and Paule [19] Paule Strehl [20] give a unifying overview of more recent works. The method of Wilf and Zeilberger, which uses a variant of Gosper s algorithm as a subroutine, is able to produce routinely short proofs of all kinds of combinatorial ....

Wilf, H. S., and Zeilberger, D. Rational functions certify combinatorial identities. J. Amer. Math. Soc. 3 (1990), 147--158.

....X j=0 ( 1 2 ) 2 j ( 5 4 ) j (1 Gamma m Gamma n) j (m n) j (1) 2 j ( 1 4 ) j ( 1 2 m n) j ( 3 2 Gamma m Gamma n) j : To finish the solution of Propp s Problem 1, we need only evaluate the sum in Theorem 15 in the case m = n. To do this we use the Wilf Zeilberger (WZ) method [26]. Lemma 16. n Gamma1 X i=0 ( 1 2 ) 2 i ( 5 4 ) i (1 Gamma 2n) i (2n) i (1) 2 i ( 1 4 ) i ( 1 2 2n) i ( 3 2 Gamma 2n) i = 4n Gamma 1 3 : Proof. Let Q(n; i) 1 4n Gamma 1 ( 1 2 ) 2 i ( 5 4 ) i (1 Gamma 2n) i (2n) i (1) 2 i ( 1 4 ) i ( 1 2 2n) i ( 3 2 Gamma ....

H. S. Wilf and D. Zeilberger, "Rational functions certify combinatorial identities,"J. Amer. Math. Soc. 3 (1990), 147--158.

....Knuth and Patashnik (1989) provides 0747 7171 90 000000 00 03.00 0 c fl 1994 Academic Press Limited 2 George E. Andrews and P. Paule an excellent algorithmic tool for finding and proving binomial single sum identities (see also Zeilberger (1991a) 1991a) and the joint paper with H. Wilf (1990)) For instance, it succeeds in proving almost all identities listed by H. Gould (1972) Its wide range of applicability also is documented by manifold and interesting new applications it has found up to now (see for instance the corresp. references above, or, as another striking example, the ....

....by V. Strehl (1990) Remark: The part of the computer in this proof certainly is simpler than the human one: identity (2. 1) can be proved by any of the standard methods, either hypergeometric (see the references above) or those described in the books by Graham, Knuth and Patashnik (1989) or Wilf (1990). The second proof is by proving more generally bm=2c X i=0 bn=2c X j=0 i j j 2 m n Gamma 2i Gamma 2j n Gamma 2j = Gamma b m n 1 2 c Delta Gamma b m n 2 2 c Delta Gamma b m 2 c Delta Gamma b m 1 2 c Delta Gamma b n 2 c ....

Wilf, H.S., Zeilberger, D. (1990). Rational functions certify combinatorial identities. J. Amer. Math.

....j=0 ( 1 2 ) 2 j ( 5 4 ) j (1 Gamma m Gamma n) j (m n) j (1) 2 j ( 1 4 ) j ( 1 2 m n) j ( 3 2 Gamma m Gamma n) j : To finish the solution of Propp s Problem 1, we need only evaluate the sum in Theorem 15 in the case m = n. To do this we use the Wilf Zeilberger (WZ) method [27]. the electronic journal of combinatorics 6 (1999) #R16 19 Lemma 16. n Gamma1 X i=0 ( 1 2 ) 2 i ( 5 4 ) i (1 Gamma 2n) i (2n) i (1) 2 i ( 1 4 ) i ( 1 2 2n) i ( 3 2 Gamma 2n) i = 4n Gamma 1 3 : Proof. Let Q(n; i) 1 4n Gamma 1 ( 1 2 ) 2 i ( 5 4 ) i (1 Gamma 2n) i ....

H. S. Wilf and D. Zeilberger, "Rational functions certify combinatorial identities,"J. Amer. Math. Soc. 3 (1990), 147--158.

....( 34n 153n 231n 117)B 3 2 n 3n 3n 1 Type: SUP SMP(FRAC INT, Symbol) Time: 0.20 (IN) 13.71 (EV) 0.23 (OT) 14.14 sec Here I will not give any further explanations w.r.t. Zeilberger s method. The interested reader may look into the numerous articles available (e.g. 25] 26] [21], 22] 24] 23] 3] 10] 9] describing the method, the algorithms, and their applications. As a side remark: I would like to take the occasion to point out another approach, experimental in character, which can be helpful in situations similar to the one discussed here. There is now a ....

H. S. Wilf and D. Zeilberger. Rational functions certify combinatorial identities. J. Amer. Math. Soc., 3:147--158, 1990.

No context found.

H.S. Wilf, D. Zeilberger, Rational functions certify combinatorial identities, Jour. Amer. Math. Soc. 3 (1990), 147-158.

.... as (n ff) ff (ff 1) n ; ff = 2 Z; 1) n ff ; ff 2 Z (2) whenever the left hand side is defined (i.e. n ff is not a negative integer) and its reciprocal as 1 (n ff) n ff 1) Gamman ff ; ff = 2 Z; n ff 1) Gamma(n ff) ff 2 Z: 3) Wilf and Zeilberger [10] associate with (n ff) its shadow ( Gamma1) n ( Gamman Gamma ff Gamma 1) which satisfies the same first order recurrence w.r.t. n. When ff = 2 Z the shadow is just a constant factor multiple of (n ff) the constant being Gamma(sin ff) while for ff 2 Z the shadow is complementary to ....

H. S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc. 3 (1990) 147--158.

.... sums sans peine (almost) 2 2 The methods The methods that have achieved the complete solution of this class of problems are the following: Sister Celine s method [1] Gosper s algorithm [3] Zeilberger s algorithm ct ( creative telescoping ) 11] Wilf and Zeilberger s WZ method [9] . Petkovsek s algorithm Hyper [6] Here is a brief description of the scope of each of these algorithms (full descriptions are in [8] Computer programs, in Maple or Mathematica versions, that carry out each of these algorithms are available free at http: www.cis.upenn.edu #wilf AeqB.html. ....

H. S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc. 3 (1990) 147 -- 158.

.... sans peine (almost) 2 2 The methods The methods that have achieved the complete solution of this class of problems are the following: ffl Sister Celine s method [1] ffl Gosper s algorithm [3] ffl Zeilberger s algorithm ct ( creative telescoping ) 11] ffl Wilf and Zeilberger s WZ method [9] ffl Petkovsek s algorithm Hyper [6] Here is a brief description of the scope of each of these algorithms (full descriptions are in [8] Computer programs, in Maple or Mathematica versions, that carry out each of these algorithms are available free at http: www.cis.upenn.edu wilf AeqB.html. ....

H. S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc. 3 (1990) 147 -- 158.

....Submitted: April 8, 1996. Accepted: April 15, 1996 ######### Using WZ pairs we present accelerated series for computing #(3) AMS Subject Classification: Primary 05A Alf van der Poorten [P] gave a delightful account of Apery s proof [A] of the irrationality of #(3) Using WZ forms, that came from [WZ1], Doron Zeilberger [Z] embedded it in a conceptual framework. 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)andA(n, k 1) A(n, k) are both rational functions. A pair (F,G) of CF functions is a WZ pair ....

H.S. Wilf, D. Zeilberger, Rational functions certify combinatorial identities,Jour.Amer.Math.Soc.3 (1990), 147-158.

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H.S. Wilf, D. Zeilberger, Rational functions certify combinatorial identities, Jour. Amer. Math. Soc. 3 (1990), 147-158.

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H. S. Wilf and D. Zeilberger, "Rational functions certify combinatorial identities,"J. Amer. Math. Soc. 3 (1990), 147--158.

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H. S. Wilf and D. Zeilberger. Rational functions certify combinatorial identities. J. Amer. Math. Soc., 3(1):147--158, 1990.

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Wilf, H. S., Zeilberger, D. (1990a). Rational Functions Certify Combinatorial Identities. J. Amer. Math.

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Wilf, H. S., Zeilberger, D. (1990a). Rational functions certify combinatorial identities. J. Amer. Math. Soc. 3, 147--158.

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