H.S. Wilf and D. Zeilberger, An Algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities, Invent. Math., 108 (1992), 575-633.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....If T is holonomic then its definite sums w.r.t. some of the variables are still holonomic as functions of the remaining variables. If T is also hypergeometric then the holonomic recurrences satisfied by these sums can be found efficiently by means of Zeilberger s Creative Telescoping algorithm [21, 22, 19]. A hypergeometric term is proper if it can be expressed as a product of a polynomial, several factorials of linear forms with integer coefficients, their reciprocals, and exponential functions. In [20] it is proved that proper hypergeometric terms are holonomic. Wilf and Zeilberger conjectured ....

....is true of the factors of each (d Gamma 1) variate sequence T (n 1 ; n i Gamma1 ; k; n i 1 ; n d ) where k 2 N . Thus by Theorem 1, each factor of T (n) is holonomic, hence by Theorem 2, so is T (n) 2 For factorial terms, this result can be found in [17] and for proper terms in [20, 19, 14]. Wilf and Zeilberger conjectured [19, p. 585] that the converse of Theorem 3 holds as well. Taken literally, this is not true as we show in Example 6 of Section 3. However, we prove in Theorem 14 a slightly modified version of their conjecture, namely that over an algebraically closed field, ....

H. S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities, Invent. Math. 108 (1992) 575--633.

.... Gamma fn = tn where tn is a given hypergeometric sequence. Besides its obvious use for indefinite hypergeometric summation, it also plays a crucial role in the algorithms for definite hypergeometric summation, construction of annihilating recurrences, and automated verification of identities [25, 26, 23]. Therefore it is not surprising that analogous algorithms have been designed for many other settings, e.g. integration of hyperexponential functions [4] basic [24, 13, 17] and bibasic [20] hypergeometric summation. We generalize Gosper s algorithm, as well as some related ones, to the mixed ....

H. S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities, Invent. Math. 108 (1992) 575--633.

....the following section employs two packages both developed at RISC. First, we use Wegschaider s [7] MultiSum for computing re currences for multiple hypergeometric sums. This Mathematica implementation extends the multivariate version of Sister Celine s technique developed by Will and Zeilberger [8]. For sake of brevity, we shall sketch the method only briefly here. For more details, the interested reader is referred to [7] Let j = jl, j, and k = kl, kr) be vectors of variables ranging over the integers. The central concept of the Sister Celine WZ method is the computation of ....

Wilf, H.S., Zeilberger, D.: An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities. Invent. Math. 108, 575 633 (1992)

....and a clear presentation of many such identities. this is due to tools like Superseeker , gfun and Mgfun [152, 24] and Rate (which is by far the most primitive of the three, but it is the most effective in this context) For hypergeometrics this is due to the WZ machinery (see [130, 190, 194, 195, 196]) And even if you should meet a case where the WZ machinery should exhaust your computer s capacity, then there are still computer algebra packages like HYP and HYPQ or HYPERG , which make you an expert hypergeometer, as these packages comprise large parts of the present hypergeometric ....

....value or coefficient in order to determine the multiplicative constant. As I explain in Appendix A, guessing can be largely automatized. It was already mentioned in the Introduction that proving binomial (hypergeometric) identities can be done by the computer, thanks to the WZ machinery [130, 190, 194, 195, 196] (see Footnote 5) Computing the degree bound is (in most cases) so easy that no computer is needed. You may use it if you want. It is only the determination of the multiplicative constant (item 5 above) by means of a special evaluation of the determinant or the evaluation of a special ....

H. S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities, Invent. Math. 108 (1992), 575--633. (p. 3, 15)

.... implementation of q analogues of Gosper s as well as of Zeilberger s [14] fast algorithm for definite q hypergeometric summation has been carried out by the author (cf. Paule and Riese [9] and Riese [12] The original approach to definite q hypergeometric summation is due to Wilf and Zeilberger [13]. The object of this paper is to describe how the algorithm qTelescope presented in [9] a q analogue of Gosper s algorithm, generalizes to the bibasic hypergeometric case. In Section 2, the underlying theoretical background based on a bibasic extension of GFF is discussed, which leads to the ....

H.S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities, Invent. Math., 108 (1992), 575--633.

....following section employs two packages both developed at RISC. First, we use Wegschaider s [7] MultiSum for computing recurrences for multiple hypergeometric sums. This Mathematica implementation extends the multivariate version of Sister Celine s technique developed by Wilf and Zeilberger [8]. For sake of brevity, we shall sketch the method only brie y here. For more details, the interested reader is referred to [7] Let j = j 1 ; j s ) and k = k 1 ; k r ) be vectors of variables ranging over the integers. The central concept of the Sister Celine WZ method is the ....

.... a ] j= j a) SUM[j] 2= j a) Sum[1= 2m 1) fm; j 1; j ag] A COMPUTER PROOF OF A SERIES EVALUATION 5 (1 j) 1 2 (1 j) 2 (2 j) 3 2 j) 1 2 (1 j) 1 2 (2 j) 0 which, after built in simpli cation, shows that it indeed satis es the recurrence: In[8]: Simplify[ Out[8] True Remark. There is no need to make the double sum terminate with respect to n. Just sum the recurrence obtained in Out[4] over n N and observe that as N 1, the resulting terms on the right hand side all vanish, because they are just polynomials p i ( j; N; k) ....

[Article contains additional citation context not shown here]

Wilf, H.S., Zeilberger, D.: An algorithmic proof theory for hypergeometric (ordinary and \q") multisum/integral identities. Invent. Math. 108, 575-633 (1992)

....term F ph is a function of the form F ph (n; k) P (n; k) Q pp r=1 (a r n b r k c r ) Q qq s=1 (u s n v s k w s ) k ; 0.1) where P is a polynomial and is a parameter. The a s, b s, u s and v s are assumed to be specific integers that do not depend on any other parameters. As in [6], we will say that F ph is well defined at (n; k) if none of the numbers fa r n b r k c r g pp 1 is a negative integer. We will say that F ph (n; k) 0 if F ph is well defined at (n; k) and at least one of the numbers fu s n v s k w s g qq 1 is a negative integer, or P (n; k) 0. A ....

....Many thanks are due to Mr. John Noonan, Professor Herbert Wilf and Professor Doron Zeilberger for their many helpful suggestions and comments during the preparation of this paper. Typeset by A M S T E X 1 2 JOHN E. MAJEWICZ thus, the name Abel type term. A remarkable fact established in [6] is that every proper hypergeometric term F ph satisfies a nontrivial recurrence relation. It did not seem to be true for Abel type terms, since for expressions of the form (0.2) the relevant ratios F (n Gamma 1; k) F (n; k) contain bad things like n n and k k ; i.e. F (n Gamma 1; k) F ....

[Article contains additional citation context not shown here]

H. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/intergral identities, Invent. math. (

....such identities. ADVANCED DETERMINANT CALCULUS 3 this is due to tools like Superseeker 2 , gfun and Mgfun 3 [152, 24] and Rate 4 (which is by far the most primitive of the three, but it is the most effective in this context) For hypergeometrics this is due to the WZ machinery 5 (see [130, 190, 194, 195, 196]) And even if you should meet a case where the WZ machinery should exhaust your computer s capacity, then there are still computer algebra packages like HYP and HYPQ 6 , or HYPERG 7 , which make you an expert hypergeometer, as these packages comprise large parts of the present hypergeometric ....

....value or coefficient in order to determine the multiplicative constant. As I explain in Appendix A, guessing can be largely automatized. It was already mentioned in the Introduction that proving binomial (hypergeometric) identities can be done by the computer, thanks to the WZ machinery [130, 190, 194, 195, 196] (see Footnote 5) Computing the degree bound is (in most cases) so easy that no computer is needed. You may use it if you want. It is only the determination of the multiplicative constant (item 5 above) by means of a special evaluation of the determinant or the evaluation of a special ....

H. S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities, Invent. Math. 108 (1992), 575--633. (p. 3, 15)

....2b 0 1; 0; a 0 b 0 1; a 0 ) for integers a 0 and b 0 . B) By [5, Theorem 1, 1.3) 2a 0 ; 2b 0 ; 1; a 0 b 0 ; a 0 ) for integers a 0 and b 0 . 9 The rst (theoretical) algorithm for proving multisum identities automatically was given by Wilf and Zeilberger [22]. A considerable enhancement and speedup was accomplished by Wegschaider [21] who combined the ideas of Wilf and Zeilberger with ideas of Verbaeten [20] Wegschaider s Mathematica implementation is available from http: www.risc.uni linz.ac.at research combinat risc software. 10 A Maple ....

H. S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and \q") multisum/integral identities, Invent. Math. 108 (1992), 575-633.

....be read, understood and verified by a human being. This paper suggests a possible approach to finding such a new proof of the Four Colour Theorem via a new reformulation of the 4CC. The idea of the suggested approach is to try to take advantage of the recent impressive progress (see, for example, [15, 19]) in computer search for closed forms of sums of the form X c 1 : X cq E(a 1 ; a p ; c 1 ; c q ) 1) in the case when all the ratios E(a 1 ; a p ; c 1 ; c r Gamma1 ; c r 1; c r 1 ; c q ) E(a 1 ; a p ; c 1 ; c r Gamma1 ; c r ; c ....

H. S. Wilf, D. Zeilberger. An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities, Invent. Math. 108:575--633 (1992). 23

.... relation (as it is done in Theorem 1 below) the existential quanti ers can, under some conditions, be replaced by summation (as in 2 Corollary 1 below) In its turn, recently we witness a spectacular progress in computer search of closed forms for sums of products of binomial coecients (see [15, 18] for accounts on this approach and for example of non trivial results obtained this way) This use of computers has the following nice feature: even if it took several CPU hours to nd a closed form, the veri cation of the found identity can be acceptible for a human being, which opens new ways to ....

....to natural bounds : if r v, s w(v r) or t wr, then the corresponding binomial coecient is equal to zero. Remark 4. Besides the above mentioned natural upper bounds for variables there is another important property which gives hope for successful application of summation technique from [15, 18], namely, the summation variables r, s and t come only linear. Acknowledgement This work was made possible by both the funding of the University of Metz and the help of Metz University Institute of Technology, giving the best conditions for both authors to meet and work together. Accordingly, ....

Wilf H. S., Zeilberger D., An algorithmic proof theory for hypergeometric (ordinary and \q") multisum/integral identities, Invent. Math. 108, 575{ 633, (1992).

.... for definite hypergeometric summation, construction of annihilating recurrences, and automated verification of identities [17, 18] Therefore it is not surprising that analogous algorithms have been designed for many other settings, e.g. integration of hyperexponential functions [4] basic [16, 15] and bibasic hypergeometric summation [13] We present in Section 6 an analogue of Gosper s algorithm for the multibasic and mixed hypergeometric case. Our algorithm m m Gosper 1 is a common generalization of algorithms presented in [8, 16, 13] Sections 2 and 3 give the required algebraic and ....

H. S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities, Invent. Math. 108 (1992) 575--633.

....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, An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities, Inventiones math., 108 (1992), 575--633.

....or solve. ffl If the resulting antidifference is complicated, it might be useful to avoid any simplifications; this can be achieved via simplify=false. By default the antidifference is factorized. If you want to apply simpfunc to the result instead, use simplify=simpfunc. 3. q Zeilberger Wilf and Zeilberger (1992) showed that Zeilberger s algorithm can be easily carried over to the q case (see also the description by Koornwinder (1993) The q Zeilberger algorithm tries to derive a recurrence equation for the definite sum S(n) S(n) bn X k=an F(n; k) 3.1) 8 H. B oing and W. Koepf where F(n; k) is ....

Wilf, H. S., Zeilberger, D. (1992). An algorithmic proof theory for hypergeometric (ordinary and `q') multisum/integral identities. Inventiones mathematicae, 108:575--633.

....with correct initial conditions. In view of (14) below, this is not too dicult, but technical (see [1] for details) Schur proved that both dm and e m satisfy the recurrence (13) c m 2 = c m 1 q m c m ; m 0: Nowadays, this identity is proved automatically by invoking the q WZ algorithm [7] and this leads to the rst purely automatic elementary proof of the Rogers Ramanujan identity [6] In view of this recurrence, dm and e m are nothing but q analogues of the Fibonacci numbers. It turns out that a generalization of the Cassini identity, namely Fm 1 Fm k Fm k 1 Fm = 1) m F k ; ....

Wilf (Herbert S.) and Zeilberger (Doron). { An algorithmic proof theory for hypergeometric (ordinary and \q") multisum/integral identities. Inventiones Mathematicae, vol. 108, n 3, 1992, pp. 575-633.

....the four color theorem [5,6] 3 . Note that the year 1976 is considered to have followed shortly after the dawn of history within the theoretical computer science community. Since that time, the method has been used to prove results in real analysis [9] dynamical systems [10] and combinatorics [11], as well as in computer science [12 17] We have answered the question What is the computational method . In the next section, we hope to tackle the question of Why . In Section 3, we illustrate the method with a simple example. In Section 4, to satisfy the more demanding reader, we present a ....

H. S. Wilf, D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and \q") multisum/integral identities, Inventiones Mathematic108 (1992) 575-633.

....theoretical background. For this, and many other exciting applications, I refer the reader to the work of Doron Zeilberger (see e.g. 12] to cite just one important article) to whom we owe most of this very recent progress, and to the comprehensive presentation by D. Zeilberger and H. Wilf in [11]. The holonomic aspect suggests that, instead of dealing with binomial sums directly, we should instead take the recurrences they satisfy as our computational objects. A nice contribution in this direction is already contained in the Maple share library, the GFUN package, which I will briefly ....

....of first degree in N . Operator and witnessing function can be computed efficiently in the case of single sums through a suitably extended version of Gosper s classical algorithm (see [12] and [13] for the Maple program) for multiple sums the method of undetermined coefficients can be used (see [11]) Maple procedures are available from D. Zeilberger (Philadelphia) call zeilberg euclid.math.temple.edu) and T. Koornwinder (Amsterdam) call thk fwi.uva.nl) together with a careful description of the method. We can now define a couple of simple Maple functions which automatize the task of ....

Herbert S. Wilf and Doron Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities, Inventiones Math., 108:575-633, 1992.

....solution f = Gamma1= qx) Hence y n = Gammab n =q n 1 satisfies (28) and P n Gamma1 j=0 b j = yn Gamma y 0 = 1 Gamma (q; q) n ) q. 2 7 Applications to q hypergeometric summation It is well known that Zeilberger s fast algorithm [18] or the more general WZmachinery described in [17], does not always deliver a representing difference equation of minimal order for the given sum. For instance, as pointed out in [11] one can prove that the Zeilberger recurrence for the sum expression on the left hand side of n X k=0 ( Gamma1) k n k d k n = Gammad) n (29) ....

H. S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities, Invent. Math. 108 (1992) 575--633.

....If T is holonomic then its definite sums w.r.t. some of the variables are still holonomic as functions of the remaining variables. If T is also hypergeometric then the holonomic recurrences satisfied by these sums can be found efficiently by means of Zeilberger s Creative Telescoping algorithm [13, 14, 11]. A function T (n 1 ; n d ) is a proper hypergeometric term if it can be expressed as a product of a polynomial with several factorials of linear forms with integer coefficients, their reciprocals, and exponential functions. In [12] it is proved that proper hypergeometric terms are ....

....that each univariate sequence f i (k) f(i; k) and f j (n) f(n; j) satisfies a recurrence with polynomial coefficients. Thus by Theorem 1, each factor of T (n; k) is holonomic, hence by Theorem 2, so is T (n; k) 2 For factorial terms, this result can be found in [9] and for proper terms in [12, 11, 7]. Example 2 Consider the sequence T (n; k) 8 : n Gamma 2k) n 2k; 3 k ; n = 2k; Gamma1) n Gamma2k (2k Gamman Gamma1) n 2k: Then T (n 1; k) a (n Gamma 2k 1)T (n; k) and (n Gamma 2k) n Gamma 2k Gamma 1)T (n; k 1) a T (n; k) so T is hypergeometric. It is ....

H. S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities, Invent. Math. 108 (1992) 575--633.

....concerning the enumeration of alternating sign matrices. There are many variations and refinements listed in [Stanl] R] MRR2,3] I am sure that the method of this paper should be capable of proving all of them. It is also possible that the present method of proof, combined with the multi WZ method[WZ], could be used to prove a stronger conjecture of [MRR1,2] conj. 3 of [Stanl] directly, in which case the present paper would also furnish an alternative proof of Andrews s[A2] TSSCPP theorem. A MORE GENERAL, AND HENCE EASIER, CONJECTURE The first step, already undertaken in [MRR2 3] is to ....

H.S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities, Invent. Math. 108(1992), 575-633.

....that it also can be used in a non obvious and nontrivial way for definite hypergeometric summation. For instance, for verifying or finding binomial identities automatically , finding recurrence operators annihilating hypergeometric sums etc. A generalization of that approach can be found in Wilf and Zeilberger (1992), which is also an excellent source for examples and further references. Maple versions of Zeilberger s algorithm have been written by Zeilberger (1991) and Koornwinder (1993) The authors of this paper implemented the algorithms of Gosper and Zeilberger in the Mathematica system (see also Paule ....

....in k (and n) Given the order d, the coefficient polynomials as well as the hypergeometric sequence g n;k are manufactured by Gosper s algorithm. In general, d is not known in advance but an upper bound can be computed. For proofs and more details of the method see e.g. Zeilberger (1990b) or Wilf Zeilberger (1992). For tutorials see Zeilberger (1993) a reprinted version is contained in this issue, or Wilf (1993) The latter also discusses the historical roots of the method referring to Sister Celine s technique, see e.g. Fasenmyer (1949) or to the work of Verbaeten, see e.g. Verbaeten (1974) Now summing ....

[Article contains additional citation context not shown here]

Wilf, H.S., Zeilberger, D. (1992). An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities. Invent. Math. 108, 575--633.

No context found.

H.S. Wilf and D. Zeilberger, An Algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities, Invent. Math., 108 (1992), 575-633.

No context found.

H. Wilf, D. Zeilberger (1992). An Algorithmic Proof Theory for Hypergeometric (ordinary and \q") Multisum/Integral Identities, Inventiones Mathematicae, 108, 575-633.

No context found.

H. S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and \q") multisum/integral identities, Invent. Math. 108 (1992), 575-633.

No context found.

H.S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and \q") multisum/integral identities, Invent. Math., 108 (1992), 575-633.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC