K. Wegschaider, Computer generated proofs of binomial multi-sum identities, diploma thesis, Johannes Kepler University, Linz, Austria, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... the work of Sister Mary Celine Fasenmyer, who showed how recurrences for certain polynomial sequences could be found algorithmically (see [15] and references therein) Proof: To prove that , we used a computer algebra package that generates computer proofs of hypergeometric multisum identities [16]. The package successfully provided a recurrence relation of the form (8) that is satisfied by the formulas for both and . Furthermore, it can be easily verified (by hand calculation or a computer algebra system) that and for . This means that under the initial conditions of and , running the ....

K. Wegschaider, "Computer generated proofs of binomial multi-sum identities," Diploma thesis, RISC, J. Kepler Univ., Linz, Austria, May 1997.

....imaginary part of the integral (2) is equal to the right hand side of (1) divided by 2r. The real part of the integral (2) may be found in standard tables. 2. THE PACKAGES The computer proof shown in 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 ....

....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 recurrences for multiple sums Ek f(J, k) where f(j, k) is hypergeometric in all of its arguments. For this, one first computes a ....

[Article contains additional citation context not shown here]

Wegschaider, K.: Computer generated proofs of binomial multi-sum identities. Diploma thesis, RISC, J. Kepler University, Linz, Austria (1997)

....the imaginary part of the integral (2) is equal to 2 times the right hand side of (1) The real part of the integral (2) may be found in standard tables. 2. The Packages The computer proof shown in the 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 ....

....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 computation of recurrences for multiple sums k f(j; k) where f(j; k) is hypergeometric in all of its arguments. For this, ....

[Article contains additional citation context not shown here]

Wegschaider, K.: Computer generated proofs of binomial multi-sum identities. Diploma thesis, RISC, J. Kepler University, Linz, Austria (1997)

....i j # (3m 2) i j) 2m 2i 2j 1) 3i 3j 2) m (4m 1) # 3(i n) 2 i n ## is zero if m nand is # 6n 4 2n 2 # 2 # 4n 3 2n 2 # if m = n. These types of identities can be proved automatically using a tool due to Wilf and Zeilberger: see for example, Kurt Wegschaider s thesis [12] for a comprehensive treatment. We rewrite the double sum, and equivalently show that for m n(below the diagonal) n # i=0 n i # j=0 ( 1) i j (3j) i j) 2 3i 3m) 1 2i 2j 2n) j (1 2j) 2 3i 3j) i m) 2 2i 2m) i j n) i j n) 0 (31) and for n = m (on the ....

K. Wegschaider. Computer Generated Proofs of Binomial Multi-sum Identities, Ph.D. thesis, Research Institute for Symbolic Computation, Institut fur Mathematik, Johannes Kepler Universitat, Linz, May 1997.

....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 implementation written by Doron Zeilberger is available from ....

K. Wegschaider, Computer generated proofs of binomial multi-sum identities, diploma thesis, Johannes Kepler University, Linz, Austria, 1997; available from http://www.risc.uni-linz.ac.at/research/combinat/risc/publications.

.... (3m 2) i j) 2m 2i 2j 1) 3i 3j 2) m (4m 1) 3(i n) 2 i n is zero if m n and is 6n 4 2n 2 2 4n 3 2n 2 if m = n. These types of identities can be proved automatically using a tool due to Wilf and Zeilberger: see for example, Kurt Wegschaider s thesis [12] for a comprehensive treatment. We rewrite the double sum, and equivalently show that for m n (below the diagonal) n X i=0 n i X j=0 ( 1) i j (3j) i j) 2 3i 3m) 1 2i 2j 2n) j (1 2j) 2 3i 3j) i m) 2 2i 2m) i j n) i j n) 0 (31) and for n = m (on ....

K. Wegschaider. Computer Generated Proofs of Binomial Multi-sum Identities, Ph.D. thesis, Research Institute for Symbolic Computation, Institut fur Mathematik, Johannes Kepler Universitat, Linz, May 1997.

....y here. For more details the reader is referred to the forthcoming article [7] The implementation is based on the method of k free recurrences, also known as Sister Celine s technique (developed by Wilf and Zeilberger [10] For reasons of eciency we also incorporated ideas from Wegschaider s [9] package MultiSum 2 for ordinary hypergeometric summation. Let n = n 1 ; n s ) and k = k 1 ; k r ) be vectors of variables ranging over the integers. The central concept of (the q version of) Sister Celine s technique is the computation of recurrences for multiple sums P k F ....

....sets to the q case, leading to more satisfactory results. The underlying existence theory was originally introduced by Verbaeten [8] for single sums in the q = 1 case. Since it is based on arguments from plane geometry, there is no direct generalization to multisums. Nevertheless, as Wegschaider [9] pointed out, P maximal structure sets can be computed also in this situation: the idea is to start with a small rectangular structure set and then to add all those points that do not increase the degree of the polynomial on the left hand side of (2.3) This way the number of equations in the ....

[Article contains additional citation context not shown here]

K. Wegschaider, Computer generated proofs of binomial multi-sum identities, Diploma thesis, RISC, J. Kepler Universitat, Linz, Austria, 1997.

....in these examples, we have to point out that this usage of Omega = manipulatorics is non algorithmic. Today we already have computer programs for the automatic treatment of hypergeometric multisums. For instance, the WZ engine of of Wilf and Zeilberger [23] has been fine tuned by Wegschaider [21]. A more general engine, based on Zeilberger s holonomic systems approach to special functions [24] has been designed by Chyzak [13] its underlying mechanism is elimination via Grobner bases methods for noncommutative operator algebras. These computer algebra packages are remarkably powerful ....

....mechanism is elimination via Grobner bases methods for noncommutative operator algebras. These computer algebra packages are remarkably powerful for various applications. For instance, see Chyzak s computer proof [13] of Calkin s identity (1. 3) or the proofs supplied by Wegschaider s package [21] including a fully automatic proof of (1.6) But still one can observe certain complexity limitations; more detailed remarks on this aspect are to find in Section 5. Therefore manipulation methods like the Omega = calculus that we are going to introduce still remain valuable tools in practical ....

K. Wegschaider, Computer Generated Proofs of Binomial Multi-Sum Identities, Diploma Thesis, RISC, J. Kepler University Linz, 1997. Available via: http://www.risc.uni-linz.ac.at/research/combinat/ risc/.

No context found.

K. Wegschaider, Computer generated proofs of binomial multi-sum identities, diploma thesis, Johannes Kepler University, Linz, Austria, 1997.

No context found.

K. Wegschaider, Computer generated proofs of binomial multi-sum identities, Diploma thesis, RISC, J. Kepler Universitat Linz, Austria, 1997.

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