P. Paule, M. Schorn (1995). A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coecient Identities. J. Symb. Comput. 20, 673-698.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....formulas in form of rules. All the available summations (with references) are listed and displayed in the manual. Besides, there is the rule SListe which for a hypergeometric series gives a list of applicable summations. First we ask which summations might be (directly) applicable to our sum. In[4]: 3 .SListe Is N a nonnegative integer [y n] y Be sure to apply FOrdne before using the following information S2101 (1 K M) S2103 (1 K M) K K Out[4] ff g, f gg (1) 1) K K There are two of them. If we want to know how they look like and what ....

....series gives a list of applicable summations. First we ask which summations might be (directly) applicable to our sum. In[4] 3 . SListe Is N a nonnegative integer [y n] y Be sure to apply FOrdne before using the following information S2101 (1 K M) S2103 (1 K M) K K Out[4]= ff g, f gg (1) 1) K K There are two of them. If we want to know how they look like and what they are we could consult the manual, or display the information on the screen as shown below. In[5] Sgl2101 Do you want to set values for the equation [y n] n ....

[Article contains additional citation context not shown here]

P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities, J. Symbolic Comput. (to appear).

....(2n Gamma i 2) 2i Gamma2 (2n Gamma 2i 1) i Summation of the relation (2.2) from i = 0 to i = n, little rearrangement, and division by n on both sides, leads to the recurrence (2n 1) S(n 1) Gamma 3 (6n Gamma 1) 6n 1) S(n) 0 for the sum in (2. 1) Paule and Schorn s [13] Mathematica implementation of the Gosper Zeilberger algorithm, which is the one we used, gives this recurrence directly. Since S(1) 1, and since the right hand side of (2.1) satisfies the same recurrence, equation (2.1) is proved, and, thus, the Corollary also. Proof of Corollary 4. First ....

P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities, J. Symbol. Comp. 20 (1995), 673--698.

....this is not the case, the situation is much more di#cult, and it is therefore quite remarkable and non obvious that Zeilberger s method is just a clever application of Gosper s algorithm. 4 REDUCE OPERATOR GOSPER 4 Our implementation is mainly based on [3] and [2] More examples can be found in [5], 7] 8] and [9] many of which are contained in the test file zeilberg.tst. 4 REDUCE operator GOSPER The ZEILBERG package must be loaded by: 1: load zeilberg; The gosper operator is an implementation of the Gosper algorithm. gosper(a,k) determines a closed form antidi#erence. If it does ....

Paule, P. and Schorn, M.: A Mathematica version of Zeilberger's algorithm for proving binomial coe#cient identities. J. Symbolic Computation, 1994, to appear.

.... 1) 1 2 ) e (n e) 3n Gamma e Gamma 1) 3 2 Gamma 2n) e = 2 n Gamma1 n (n Gamma 1) 6n Gamma 3) 3n) 4n Gamma 3) Let us denote the sum by S(n) Then an application of the Gosper Zeilberger algorithm [17, 22, 23] we used the Mathematica implementation by Paule and Schorn [16]) yields the relation 2n(2n 1) 6n Gamma 1) 6n 1)S(n) Gamma (3n 1) 3n 2) 4n Gamma 1) 4n 1)S(n 1) 0; which easily proves the claimed summation by an induction on n. Outline of proof of Theorem 4. From MacMahon s formula (1.1) for the total number of rhombus tilings together with ....

P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities , J. Symbol. Comp. 20 (1995), 673--698.

....of memory. However, as we already mentioned in the Introduction, in some cases formulas in form of single hypergeometric sums are available. If one is in such a case then one would proceed as in the above paragraph, but one would replace the multisum algorithm by Zeilberger s algorithm 10 (see [15, 16, 23, 24]) The advantage is that, in contrast to the multisum algorithm, Zeilberger s algorithm is very ecient. At any rate, in any case that I looked at in connection with our problem, the Zeilberger algorithm was successful. That is to say, if I am allowed to somewhat overstate it, whenever one is in a ....

P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial coecient identities, J. Symbol. Comp. 20 (1995), 673-698.

....is not the case, the situation is much more difficult, and it is therefore quite remarkable and non obvious that Zeilberger s method is just a clever application of Gosper s algorithm. 4 REDUCE OPERATOR GOSPER 4 Our implementation is mainly based on [3] and [2] More examples can be found in [5], 7] 8] and [9] many of which are contained in the test file zeilberg.tst. 4 REDUCE operator GOSPER The ZEILBERG package must be loaded by: 1: load zeilberg; The gosper operator is an implementation of the Gosper algorithm. ffl gosper(a,k) determines a closed form antidifference. If it ....

Paule, P. and Schorn, M.: A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. J. Symbolic Computation, 1994, to appear.

....[22] x 2.4) and can be found in ( 7] x 10.23, 6) However, these relations nowhere appeared systematically. The important issue of our presentation is its algorithmic content: All given representations can be calculated by a computer algebra system, e.g. by implementations in Mathematica ([23] and [15] Reduce [14] and Maple [13] Note that in our Mathematica implementation [15] all the partial algorithms mentioned are applied completely automatically. 5 2 Application to Feynman diagrams Zeilberger s algorithm can be applied to find hypergeometric identities (see e.g. 15] and ....

.... = n Gamma1 X k=j n (n Gamma k) k (k Gamma j 1) j (n Gamma j 1) j 1 (ff 1 k) n Gammak = A(n; j) 45) The last identity can easily be proved by Zeilberger s algorithm: It turns out that both sides of (45) satisfy the inhomogeneous first order recurrence equation (see e.g. [23]) a(n 1; j) Gamma a(n; j) A(n 1; j) Gamma A(n; j) 1 1 ff n of the function, having the same initial value a(j 1; j) A(j 1; j) 1 ff 1 j . Acknowledgement I like to thank Prof. Peter Deuflhard for his support, and encouragement to work on the given subject, and for his ....

Paule, P. and Schorn, M.: A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. J. Symbolic Computation, to appear 1995.

....and Burge [2] For the combinatorial context in which determinants such as (1) arise, see the article [6] of Robbins. Zeilberger s algorithm, which we used to prove this result, is in [7] The particular implementation of it that we used, due to Peter Paule and Markus Schorn, is described in [4]. A discussion of the whole field of computer proofs of identities is in the book [5] Though the idea was beautiful and simple, its execution was anything but. Indeed, the proofs of the triangularity of the product and of the nature of its diagonal entries required an argument that proved, by ....

P. Paule and M. Schorn, A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coefficient Identities, RISC-Linz Report Series 93-36, J. Kepler University, Linz (1993).

.... and their generating functions is the main topic of this package ( 8] 15] extending the previous package PowerSeries [12] Moreover the package automatically finds differential and recurrence equations ( 13] 14] for expressions and for sums (the latter using Zeilberger s algorithm ( 23] [18], 13] As an application the fast computation of polynomial approximations of solutions of linear differential equations with polynomial coefficients is presented. This is the asymptotically fastest known algorithm for series computations, and it is much faster than Mathematica s builtin Series ....

....Log[1 x] Out[17] x By this reason we use the (non evaluating) name hypergeometricPFQ instead. Note that the procedure PowerSeries can handle the more general case of Laurent Puiseux series representations (p = 1; 2; f(x) 1 X k= Gamma1 a k x k=p ; for example In[18]: PS[Sin[Sqrt[x] x] k 1 2 k ( 1) x Out[18] sum[ k, 0, Infinity] 1 2 k) 3 Recurrence Equations Similarly to the sum and product algorithms for holonomic differential equations there are simple sum and product algorithms for holonomic recurrence equations. These compute ....

[Article contains additional citation context not shown here]

Paule, P. and Schorn, M.: A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. J. Symbolic Computation 20, 1995, 673--698.

....formulas in form of rules. All the available summations (with references) are listed and displayed in the manual. Besides, there is the rule SListe which for a hypergeometric series gives a list of applicable summations. First we ask which summations might be (directly) applicable to our sum. In[4]: 3 .SListe Is N a nonnegative integer [y n] y Be sure to apply FOrdne before using the following information S2101 (1 K M) S2103 (1 K M) K K Out[4] ff g, f gg (1) 1) K K There are two of them. If we want to know how they look like and what ....

....series gives a list of applicable summations. First we ask which summations might be (directly) applicable to our sum. In[4] 3 . SListe Is N a nonnegative integer [y n] y Be sure to apply FOrdne before using the following information S2101 (1 K M) S2103 (1 K M) K K Out[4]= ff g, f gg (1) 1) K K There are two of them. If we want to know how they look like and what they are we could consult the manual, or display the information on the screen as shown below. In[5] Sgl2101 Do you want to set values for the equation [y n] n ....

[Article contains additional citation context not shown here]

P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities, J. Symbolic Comput. (to appear).

....system Mathematica and developed a user interface that dispenses the user from working explicitly with difference fields. Instead, the user can handle all summation problems in terms of sums and products. This algorithm cannot only deal with series of hypergeometric terms, like Gosper s algorithm [Gos78, PS95], series with q hypergeometric terms, like [PR97] or holonomic series, like Chyzak s algorithm [CS98] but with series of terms where for example the harmonic numbers can appear in the denominator (see section 2.4) In some cases appropriate difference field extensions are necessary in order to ....

P. Paule and M. Schorn. A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. J. Symbolic Computation, 20:673-- 698, 1995.

....g(i) If such an g(i) was found we would have 1 X i=0 f(i) 1 X i=0 (g(i 1) g(i) lim i 1 (g(i) g(0) lim i 1 g(i) exists in our cases, since f(i) 0 for all but a nite number of i and therefore g(i) is nally constant. A computer implementation of Gosper s algorithm [12] prints out ( 1) j ( a b k) i j 2 1 2 a 2 k 2 i ( 1 b k) i j k) 2(1) 1 a i j 1 a 2 b 2 k i 1 to be a suitable g(i) for our f(i) One may check the identity g(i 1) g(i) f(i) by dividing the left hand side by the right hand side and simplifying the resulting ....

P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial identities, J. Symbol. Comp. 20 (1995), 673-698.

....and Burge [2] For the combinatorial context in which determinants such as (1) arise, see the article [6] of Robbins. Zeilberger s algorithm, which we used to prove this result, is in [7] The particular implementation of it that we used, due to Peter Paule and Markus Schorn, is described in [4]. A discussion of the whole field of computer proofs of identities is in the book [5] Though the idea was beautiful and simple, its execution was anything but. Indeed, the proofs of the triangularity of the product and of the nature of its diagonal entries required an argument that proved, by ....

P. Paule and M. Schorn, A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coefficient Identities, RISC-Linz Report Series 93-36, J. Kepler University, Linz (1993).

....Gosper [9] or Graham, Knuth and Patashnik [10] is implemented in most computer algebra systems. Extensions to Zeilberger s algorithm have been done by Zeilberger [24] and Koornwinder [11] in Maple. A very powerful Mathematica version of Zeilberger s algorithm has been written by Paule and Schorn [17]. However, implementations of q analogues are anything but widespread. Up to now there exist only two Maple versions: one, being on a quite rudimentary level, written by Zeilberger, and one by Koornwinder [11] We will come back to the latter one in the following subsection. Installation The ....

P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities, RISC-Linz Report Series, 95-10 (1995), to appear in J. Symb. Computation.

.... function multiple of the original summand f n;k with undetermined i , can be used to compute both the polynomials i and the sequence g n;k from (1) Several implementations of Zeilberger s algorithm have been carried out; the most prominent ones are due to Koornwinder [8] Paule and Schorn [11], and Zeilberger (see Petkov sek, Wilf and Zeilberger [13] Since our methods will be shown to work also in the q hypergeometric universe, we brie y comment on the underlying theory. A sequence f n;k with values in F(q) is called q hypergeometric if the quotients fn 1;k f n;k and f n;k 1 f ....

P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial coecient identities, J. Symbolic Comput., 20 (1995), 673-698. 9

.... the more general q case, for which a Mathematica version has already been developed at RISC (see [7] Moreover we remark that at RISC a Mathematica version of Zeilberger s fast algorithm for the ordinary case q = 1, considered in this report, has already been implemented by Paule and Schorn [6]. Both packages can be downloaded from the home page http: www.risc.uni linz.ac.at research combinat risc . 3 Chapter 2 Some Theory We now introduce the basic de nitions of the theory of de nite and inde nite hypergeometric summation and some fundamental results of the theory which are ....

.... 5 Performance First of all we must point out that this implementation is only the rst attempt to incorporate Zeilberger s fast algorithm into the Macsyma computer algebra system and therefore it is very far from being competitive with the best existing optimized implementations (for example see [6]) No large scale test has been carried out to assess the speed of this implementation. The only test that has been run on this implementation is the collection of sequences contained in the les testGosper.macsyma and testZeilberger.macsyma, among which there are some powers of the binomial ....

[Article contains additional citation context not shown here]

Paule, P., Schorn, M. A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coe cient Identities J. Symbolic Computation, 20, 1995, 673-698.

....of Zeilberger s algorithm ct corresponding to d =0,andtod = 1 with given closed form evaluation, respectively (d being the order of the resulting recurrence) Besides our own implementations, we used the outstanding implementation of Zeilberger s algorithm in Mathematica by P. Paule and M. Schorn [5], which excels especially when the resulting recurrence is not homogeneous. Many of the problems were solved completely automatically, while others required a little human help. For example, in several sums that involve the floor function we humans carried out the replacement # k F (n, k, ....

P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial coe#cient identities, J. Symb. Comput. 20 (1995) 673--698.

....2 (2n Gamma i 2) 2i Gamma2 (2n Gamma 2i 1) i 2 : Summation of the relation (2.2) from i = 0 to i = n, little rearrangement, and division by n on both sides, leads to the recurrence (2n 1) 2 S(n 1) Gamma 3 (6n Gamma 1) 6n 1) S(n) 0 for the sum in (2. 1) Paule and Schorn s [13] Mathematica implementation of the Gosper Zeilberger algorithm, which is the one we used, gives this recurrence directly. Since S(1) 1, and since the right hand side of (2.1) satisfies the same recurrence, equation (2.1) is proved, and, thus, the Corollary also. Proof of Corollary 4. First ....

P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities, J. Symbol. Comp. 20 (1995), 673--698.

....(2.3) from i = 0 to i = n Gamma 1, and little rearrangement, leads to the recurrence 6n 2 (n 2) 6n 1) 6n 5) S(n) Gamma 6(n 1) 2n Gamma 1) 2 (3n 1) 3n 2) S(n 1) n 2) 2n Gamma 1) 2 (36n 3 60n 2 29n 3) 2(n 1) for the sum in (2. 2) Paule and Schorn s [35] Mathematica implementation of the Gosper Zeilberger algorithm, which is the one we used, gives this recurrence directly. Since S(1) 1, and since the right hand side of (2.2) satisfies the same recurrence, equation (2.2) is proved. The procedure in the other two cases is analogous. The ....

P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities, J. Symbol. Comp. 20 (1995), 673--698.

....algorithm ct corresponding to d = 0, and to d = 1 with given closed form evaluation, respectively (d being the order of the resulting recurrence) Besides our own implementations, we used the outstanding implementation of Zeilberger s algorithm in Mathematica by P. Paule and M. Schorn [5], which excels especially when the resulting recurrence is not homogeneous. Many of the problems were solved completely automatically, while others required a little human help. For example, in several sums that involve the floor function we humans carried out the replacement X k F (n; k; ....

P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities, J. Symb. Comput. 20 (1995) 673--698.

....m; k) F (n; k) and F (n; k l) F (n; k) are rational functions with respect to n and k for a certain pair (m; l) 2 IN 2 . For example, by (3) 4) this is valid for F (n; k) Gamma n k Delta 2 with m = l = 1. A modification [21] of the (fast) Zeilberger algorithm ( 41] see also [25] and [29]) returns a holonomic recurrence equation valid for s(n) Zeilberger s algorithm is based on a decision procedure for indefinite summation due to Gosper [16] In our example case, Zeilberger s algorithm finds the holonomic recurrence equation (1 n) s(n 1) 2(1 2n) s(n) for s(n) P k2ZZ ....

....(z Gamma 1) z V (ff 2; fi; fl) and analogous recurrence equations with respect to the variables fi and fl (see [22] These, in particular, can be used for numerical purposes. Note that for the application of Zeilberger s algorithm our Mathematica program uses the Paule Schorn implementation [29]. For the current example, the output is given by In[11] HolonomicRE[ 1) alpha beta gamma) Gamma[alpha beta gamma d] Gamma[d 2 gamma] Gamma[alpha gamma d 2] Gamma[beta gamma d 2] Gamma[alpha] Gamma[beta] Gamma[d 2] Gamma[alpha beta 2 gamma d] M(alpha beta gamma d) Hypergeometric2F1[ ....

[Article contains additional citation context not shown here]

Paule, P. and Schorn, M.: A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. J. Symbolic Computation, 1995, to appear.

....antiderivative is known. If this is not the case, the situation is much more difficult, and it is therefore quite remarkable and non obvious that Zeilberger s method is just a clever application of Gosper s algorithm. Our implementation is mainly based on [2] Many more examples can be found in [4], 6] 7] and [8] most of which are contained in the test file zeilberg.tst. 3 REDUCE operator GOSPER The ZEILBERG package must be loaded by: 1: load zeilberg; The gosper operator is an implementation of the Gosper algorithm. ffl gosper(f,k) determines a closed form antidifference. If it does ....

Paule, P. and Schorn, M.: A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. J. Symbolic Computation, 1994, to appear.

....2 (2n Gamma i 2) 2i Gamma2 (2n Gamma 2i 1) i 2 : Summation of the relation (2.2) from i = 0 to i = n, little rearrangement, and division by n on both sides, leads to the recurrence (2n 1) 2 S(n 1) Gamma 3 (6n Gamma 1) 6n 1) S(n) 0 for the sum in (2. 1) Paule and Schorn s [11] Mathematica implementation of the Gosper Zeilberger algorithm, which is the one we used, gives this recurrence directly. Since S(1) 1, and since the right hand side of (2.1) satisfies the same recurrence, equation (2.1) is proved, and, thus, the Corollary also. 4 M. CIUCU AND C. ....

P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities, J. Symbol. Comp. 20 (1995), 673--698.

....2 N 2 . For example, by (3) 4) this is valid for F (n; k) Gamma n k Delta 2 with m = l = 1. We assume moreover that the sums (5) have finite support, i.e. they are finite sums for each particular n 2 N. A modification [23] of the (fast) Zeilberger algorithm ( 43] see also [27] and [31]) returns a holonomic recurrence equation valid for s(n) Zeilberger s algorithm is based on a decision procedure for indefinite summation due to Gosper [17] In our example case, Zeilberger s algorithm finds the holonomic recurrence equation (1 n) s(n 1) 2(1 2n) s(n) for s(n) P k2Z ....

....(z Gamma 1) z V (ff 2; fi; fl) and analogous recurrence equations with respect to the variables fi and fl (see [24] These, in particular, can be used for numerical purposes. Note that for the application of Zeilberger s algorithm our Mathematica program uses the Paule Schorn implementation [31]. For the current example, the output is given by In[18] HolonomicRE[ 1) alpha beta gamma) Gamma[alpha beta gamma d] Gamma[d 2 gamma] Gamma[alpha gamma d 2] Gamma[beta gamma d 2] Gamma[alpha] Gamma[beta] Gamma[d 2] Gamma[alpha beta 2 gamma d] M(alpha beta gamma d) ....

[Article contains additional citation context not shown here]

Paule, P. and Schorn, M.: A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. J. Symbolic Computation, to appear (1996).

....k g(n; k) where the summand g(n; k) is hypergeometric in both variables, can be systematically treated by Zeilberger s algorithm (method of creative telescoping, see for instance [Zei90] or [PWZ96, Chapter 6] Zeilberger s algorithm has been implemented in Mathematica by P. Paule and M. Schorn [PS95]. CHAPTER 2. MY MATHEMATICA PACKAGE 52 2.7.2 DifferentialEquationPlus (DEPlus) DifferentialEquationHadamard (DEHadamard) DifferentialEquationCauchy (DECauchy) DEPlus[de 1 ,de 2 ,f[x] gives a differential equation that is satisfied by the sum of solutions of the differential equations de 1 and ....

P. Paule and M. Schorn. A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. Journal of Symbolic Computation, 20:673--698, 1995.

....heuristic approach consists in computing sufficiently many terms of the sequence and then letting the guessing mechanism of gfun find the recurrence. Since the summand is hypergeometric in both variables, it is also possible to use Zeilberger s fast algorithm to get the recurrence (see e.g. Paule Schorn 1995). We detail here how the equation is rigourously constructed via the univariate closure properties. We start from the trivial order 0 recurrence satisfied by 1= k 2 1) rec: k2 1) u(k) 1: from which we deduce the differential equation satisfied by its generating function: ....

Paule, P. and M. Schorn (1995). A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. Journal of Symbolic Computation (this issue).

No context found.

P. Paule and M. Schorn. A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. J. Symbolic Computation, 20:673--698, 1995.

....are again free of k. This recurrence now has the appropriate form for summation and immediately yields a recurrence for the whole sum. In the second part of the proof, we shall compute closed forms for the inner sums of S(1) and S(2) This will be achieved with the Paule Schorn 2 implementation [3] of Zeilberger s fast algorithm [4] which computes recurrences for hypergeometric single sums f(n, k) This is done by first determining polynomials ai(n) and a rational function r(n, k) such that d i(n) f(n i, k) r(n, k)f(n, k) i O Again, this recurrence may be summed easily, ....

Paule, P., Schorn, M.: A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. J. Symbolic Con]put. 20, 673-698 (1995)

....are again free of k. This recurrence now has the appropriate form for summation and immediately yields a recurrence for the whole sum. In the second part of the proof, we shall compute closed forms for the inner sums of S(1) and S(2) This will be achieved with the Paule Schorn implementation [3] of Zeilberger s fast algorithm [4] which computes recurrences for hypergeometric single sums f(n; k) This is done by rst determining polynomials i (n) and a rational function r(n; k) such that d i=0 i (n) f(n i; k) k (r(n; k)f(n; k) Again, this recurrence may be summed ....

....for f (j; n; k) For this, we start out by utilizing a brand new feature of MultiSum that quickly determines a small structure set. In[1] MultiSum.m MultiSum Package by Kurt Wegschaider c RISC Linz V 1. 45 (03 28 02) In[2] f = 2n 2 ) 2n 1 k) k 2 (k j n 1=2) In[3]: S = FindRecurrence[f , j, 2, fn; kg, f1; 1g, f2; 0g, WZ True, Protocol False, NumericCheck True] Candidate for structure set: Out[3] ff0; 0; 0g; f0; 1; 0g; f0; 1; 1g; f1; 0; 0g; f1; 1; 0g; f1; 1; 1g; f2; 0; 0g; f2; 1; 1gg Now we use this set to compute a symbolic solution. This ....

[Article contains additional citation context not shown here]

Paule, P., Schorn, M.: A Mathematica version of Zeilberger's algorithm for proving binomial coecient identities. J. Symbolic Comput. 20, 673-698 (1995)

....depending on the parity of m, whence the result. Remark. Let us mention that proofs of binomial summations like the Vandermonde formula or Lemma 4 can nowadays be carried out in a purely automatic fashion due to Zeilberger s summation machinery [PWZ96] see, for instance, the Mathematica package [PS95]. We are now ready to carry out the last reduction step. As opposed to Propositions 2 and 3 that relate identities between the h i and the Q i , the following proposition states an reduction between the equations in the Q i only. Proposition 5. Under the simultaneous assumption of the cases l = ....

P. Paule and M. Schorn. A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. J. Symbolic Computation, 20:673--698, 1995.

....(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) for a fixed positive integer d is of order d Gamma 1 instead of order 1 according to its hypergeometric evaluation. Here one applies ....

.... 2 n 1 3 n (q (1 q q ) 1 q q q ) n 1 2 n SUM[ 1 n] 1 q ) 1 q ) The algorithm qHyper now finds the q hypergeometric solution of this recurrence (after replacing q n by x, and SUM[n k] by Y[qk x] In[10] SUM[n k. Y[qk x] q(a. n b. xa qb; In[11]: qHyper[ Y[x] Warning: irreducible factors of degree 1 in leading coefficient; some solutions may not be found Warning: irreducible factors of degree 1 in trailing coefficient; some solutions may not be found 17 2 2 2 Out[11] x (1 q x q x ) This means that for one solution y n ....

[Article contains additional citation context not shown here]

P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities, J. Symb. Comp. 20 (1995) 673-- 698.

.... = f(1) f( Gamma1) 2 = 2 m n Gamma1 : Remark 1. We would like to mention that due to Zeilberger s summation machinery [20] today proofs of binomial summations like Vandermonde s formula or Lemma 2 can be carried out in a purely automatic fashion; see, for instance, the Mathematica package [19]. Now we are ready to carry out the last reduction step. Proposition 3. The case l = 0 of Proposition 2 is a consequence of Proposition 1 together with the cases l = 1; N Gamma 1 of Proposition 2. Proof. By L(i; j; l) we denote the double sum on the left hand side of (12) First we note ....

Peter Paule and M. Schorn. A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. J. Symbolic Computation, 20:673--698, 1995.

....(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 WZ machinery 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) for a fixed positive integer d is of order d Gamma 1 instead of order 1 according to its hypergeometric evaluation. Here one applies ....

.... 2 n 1 3 n (q (1 q q ) 1 q q q ) n 1 2 n SUM[ 1 n] 1 q ) 1 q ) The algorithm qHyper now finds the q hypergeometric solution of this recurrence (after replacing q n by x, and SUM[n k] by Y[qk x] In[10] SUM[n k. Y[qk x] q(a. n b. xa qb; In[11]: qHyper[ Y[x] Warning: irreducible factors of degree 1 in leading coefficient; some solutions may not be found 15 Warning: irreducible factors of degree 1 in trailing coefficient; some solutions may not be found 2 2 2 Out[11] x (1 q x q x ) This means that for one solution y n ....

[Article contains additional citation context not shown here]

P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities, RISC-Linz, Report Series No.95-10, to appear in J. Symb. Comp.

....Apollo 4500 workstation for obtaining the certificates (of order 2) of both sides. The qWZ certificates presented in this paper are obtained in about 60 seconds or less. A more detailed discussion of algorithmic background, performance, and examples will be given in paper by the author and Riese [19]. Acknowledgment: The author thanks Volker Strehl, Herb Wilf and Doron Zeilberger for valuable suggestions and comments. ....

P. Paule and A. Riese, A Mathematica version of Zeilberger's Algorithm for proving terminating q-hypergeometric series identities, in preparation.

....Koornwinder [1993] we felt the need to come up with another implementation qZeil, in Mathematica, which is able to deal with various important features not covered by the ones mentioned. A similar project was carried out for the ordinary hypergeometric case; see the Mathematica implementation by Paule and Schorn [1993]. 1991 Mathematics Subject Classification. Primary 33D15, 33D20, 68Q40; Secondary 05A30, 68R05. The work of the first author was supported in part by grant P7220 of the Austrian FWF. c fl0000 American Mathematical Society 0000 0000 00 1.00 .25 per page 2 Peter Paule and Axel Riese ....

....the understanding of the various program features are described in short, for instance: qWZ certification, qWZ pairs, qWZ dualization and qWZ companion identities. The section concludes with a brief comparison with Koornwinder s implementation. Section 5 deals with nontrivial applications. As in Paule and Schorn [1993], a special emphasis is put on illustrating how to apply the package to concrete problems. Besides the examples one finds in Section 4, in this section we present: an example where a computer gave the first proof of a (human) conjecture, a detailed discussion of creative symmetrizing for ....

[Article contains additional citation context not shown here]

Paule, P. and Schorn, M. [1993] A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities, RISC-Linz Report Series, 93-36, J. Kepler University, Linz, to appear in: J. Symbolic Computation.

....corresponding linear recurrences, both being of order 3 instead of order 1 in case of hypergeometric term evaluation. The recurrences, in i or n respectively, can be obtained automatically by applying Zeilberger s algorithm. For example, that one in i is obtained by the Zeilberger package by P. Paule and M. Schorn (1993), written in Mathematica, as follows. The package is available via email from Peter.Paule risc.uni linz.ac.at. Let SUM [i] n X j=0 i j j 2 4n Gamma 2i Gamma 2j 2n Gamma 2j ; 1.2) then: In[1] Zb[Binomial[i j,j]2 Binomial[4n 2i 2j,2n 2j] j,i,3] Out[1] 1 i) i ....

....recurrence relation. Here the part of the human consists in clever guessing on base of the data delivered by the computer. We want to remark that hypergeometric methods (see the references above) seemingly do not succeed applied in various standard ways. Equipped with the Zeilberger package by Paule and Schorn (1993), or that one written by Zeilberger (1991b) one has to do a so far human preprocessing step. One rewrites the double sum as a single sum, i.e. in a form, which is ready to serve as input for the proving procedure. The left side of (1.1) is equal to the coefficient of x n y n in (1 x) ....

[Article contains additional citation context not shown here]

Paule, P., Schorn, M. (1993). A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities - A description how to use it. In preparation.

....Apollo 4500 workstation for obtaining the certificates (of order 2) of both sides. The qWZ certificates presented in this paper are obtained in about 60 seconds or less. A more detailed discussion of algorithmic background, performance, and examples will be given in paper by the author and Riese [19]. Acknowledgment: The author thanks Volker Strehl, Herb Wilf and Doron Zeilberger for valuable suggestions and comments. ....

P. Paule and A. Riese, A Mathematica version of Zeilberger's Algorithm for proving terminating q-hypergeometric series identities, in preparation.

.... ( Zei90a] that is based on Gosper s algorithm ( Gos78] for indefinite hypergeometric summation (see also [PWZ96] or [GKP94] In the few cases where we make use of Zeilberger s fast algorithm in this thesis, we use the reliable and fast Mathematica implementation due to Paule and Schorn ( PS95] 1 For multiple summation no equally successful method exists. The problem is much harder to solve, and despite all efforts, the running time of the algorithms can be prohibitively high. There are two essentially different methods to compute the recurrence (1.4) the hypergeometric method ....

....a list of variables and MainVars is a single variable, but occasionally the sum involves two discrete main variables and it may be of advantage to look for a recurrence in both of them. The Summand is defined, similar to the input for the Paule Schorn implementation of Zeilberger s algorithm ( PS95] as Summand : hgterm hgterm : simpleterm or Power[ hgterm , integer ] or Times[ hgterm , hgterm ] simpleterm : rational or Binomial[ intlinpoly , intlinpoly ] or Factorial[ intlinpoly ] or Gamma[ intlinpoly ] or Power[ constrational ....

P.Paule and M. Schorn. A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. J. Symbolic Comp., 20:673--698, 1995.

No context found.

P. Paule, M. Schorn (1995). A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coecient Identities. J. Symb. Comput. 20, 673-698.

No context found.

Paule, P., and Schorn, M. A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coe- cient Identities. J. Symbolic Computation 20 (1995), 673-698.

No context found.

P. Paule, M. Schorn, 1995. A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coefficient Identities. J. Symbolic Computation, Vol. 20, pp. 673-698.

No context found.

P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities, J. Symbol. Comp. 20 (1995), 673--698.

No context found.

P. Paule and M. Schorn. A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. J. Symbolic Computation, 1995.

No context found.

Paule, P., Schorn, M. (1994). A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. J. Symbolic Computation, to appear.

No context found.

Paule, P., and Schorn, M. A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. Journal of Symbolic Computation, 1995. To appear.

No context found.

Paule, P., Schorn, M. (1995). A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. Journal of Symbolic Computation.

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