F. Chyzak, B. Salvy (1998). Non-commutative Elimination in Ore Algebras Proves Multivariate Identities. J. Symb. Comput. 26, no. 2, 187-227.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....arbitrary left submodules of Dr. Definition 2.0.3. Let N C D r be a left D subraodule. The Weyl closurEe of N, denoted CI(N) is the submodule Cl(N) a. N n o r. The question of computing the Weyl closure for finite rank ideals of D (see Definition 2.1. 2 for a definition of rank) was posed in [14] by Chyzak and Salvy, who call this question the extension contraction problem and consider it more generally for left ideals of Ore algebras. Their motivation to compute Weyl closure is for non commutative elimination and its application to symbolic integration. Namely, given a left ideal I in ....

....which vanish on the common zeroes of J. 2.2 Finite rank algorithm In this section, we provide an algorithm to compute the Weyl closure of a sub module N C D r such that the quotient module M Dr N has finite rank. This solves the extension contraction problem posed by Chyzak and Salvy in [14] for the case of the Weyl algebra. To obtain the algorithm, we first identify polynomials f such that for a given submodule N C D r, we have CI(N) D[f 1] N fq Dr. The algorithm is then a direct application of the localization algorithm of Oaku Takayama Walther [36] In the next section, we will ....

Chyzak, F., Salvy, B. (1998): Non-commutative elimination in Ore algebras proves multivariate holonomic identities. Journal of Symbolic Computation 26, 187-227.

....and presents it in a uniform way, with the added possibility of interactivity. We concentrate on special functions that are solutions of a linear di#erential or di#erence equation. These functions are called D finite or holonomic and many of their properties are algorithmically computable [9, 11, 8, 4, 5]. It turns out that this class covers about 60 of the functions described in [2] In other words, 60 of this reference can now be produced automatically with very little extra coding on top of standard computer algebra packages. Section 2 presents important properties of holonomic functions that ....

Frederic Chyzak and Bruno Salvy. Non-commutative elimination in Ore algebras proves multivariate holonomic identities. Journal of Symbolic Computation, 26(2):187--227, August 1998.

....(i.e. decomposed in first order factors) minimal annihilators. Together with the algorithm we describe its Maple implementation. It is easy to observe that P1 P4 can be considered not only in the differential case but also in the difference and the q difference cases. The concept of Ore rings ([16, 10, 11, 12, 13]) lets one design universal algorithms which can be adjusted on one or another concrete case. The algorithm and the program described below are universal and find the minimal annihilators in all cases covered by the Ore ring approach. 2 Generalities Let k be a field of characteristic zero, X an ....

F. Chyzak, B. Salvy (1996): Non-commutative elimination in Ore algebra proves multivariate holonomic identities, INRIA Research Report, No 2799.

....nice two variable functions, extraction of the diagonal is e ective and asymptotics may then be obtained. In more than two dimensions, no analytic expression for the diagonal is available [Sta99] but the diagonal is still D nite [Lip88] and a recursion for the diagonal may be e ectively derived [ChySal96], which allows the derivation of asymptotics by solving di erence equations with polynomial coecients. This has in fact been implemented [LeyTsa00] and has no problem running on a standard laptop (circa 1999) when the inputs are reasonable. The methods used in these cases, though super cially ....

Chyzak, F. and Salvy, B. (1998). Non-commutative elimination in Ore Algebras proves multivariate identities. J. Symbol. Comput. 26 187 - 227.

....method. It is known (Lipshitz 1988) that the diagonal sequence a n;n; n of a multivariate sequence with rational generating function has a generating function satisfying a linear di erential equation over rational functions. Much is known about how to compute this equation (see for example Chyzak Salvy (1998)) If one wants asymptotics on the diagonal, or in any direction where the coordinate ratios are rational numbers with small denominators, then these methods give results that are in theory at least as good as ours. The method, however, is inherently non uniform in the direction, so there is no ....

Chyzak, F. & Salvy, B. (1998). `Non-commutative elimination in Ore Algebras proves multivariate identities', J. Symbol. Comput. 26, 187-227.

....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 find a closed form to a summation problem. In many cases our implementation is able to find such ....

F. Chyzak and B. Salvy. Non-commutative elimination in ore algebras proves multivariate identities. J. Symbolic Computation, 26(2):187--227, 1998.

....can always be transformed to proving identities of sequences (see Theorem 1.4.1) we now focus our considerations on the ring of linear recurrence operators. General discussions may be found in [Ore33] BP94] or in [Li96] A thorough treatment of multivariate Ore polynomial rings is presented in [CS]. From now on we assume that oe is defined as in Example 3.3.2. Let A and B be two nonzero polynomials in (Kn ) E] with the respective degrees d and e in E. Without loss of generality we assume that d e. Let a and b be the leading coefficient (lc) of A and B, respectively. If we set Q 0 to be ....

F. Chyzak and B. Salvy. Non-commutative elimination in Ore algebras proves multivariate identities. To appear in the Journal of Symbolic Computation.

....of the form (1.4) for any holonomic function. Zeilberger used Sylvester s dialytic elimination to compute this recurrence, a method that is rather slow. The work on such annihilation ideals was extended by F. Chyzak and B. Salvy to ffi finite functions in the general setting of Ore algebras ( CS97] In this context elimination is performed by using noncommutative Grobner bases. These ffi finite functions can be handled with F. Chyzak s ( Chy94] Maple implementation Mgfun. 2 It remains to mention that M. Schorn, in the last chapter of his diploma thesis [Sch95] presents an elimination ....

F. Chyzak and B. Salvy. Non-commutative elimination in Ore algebras proves multivariate holonomic identities. Journal of Symbolic Computation, 1997. Submitted.

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Chyzak, F., Salvy, B. "Non-commutative elimination in Ore algebras proves multivariate identities ", J. Symbolic Computation, 26 (1998), 187-227.

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Chyzak, F., Salvy, B. "Non-commutative elimination in Ore algebras proves multivariate identities ", J. Symbolic Computation, 26 (1998), 187-227.

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Chyzak, F. and Salvy, B. "Non-commutative elimination in Ore algebras proves multivariate identities", J. Symbolic Computation, vol. 26 (1998), 187-227.

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Chyzak, F., Salvy, B. \Non-commutative elimination in Ore algebras proves multivariate identities ", J. Symbolic Computation, 26 (1998), 187-227.

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Chyzak, F. and Salvy, B. Non-commutative elimination in Ore algebras proves multivariate identities, J. Symbolic Computation, vol. 26 (1998), 187-227.

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Chyzak, F. and Salvy, B. "Non-commutative elimination in Ore algebras proves multivariate identities", J. Symbolic Computation, vol. 26 (1998), 187-227.

....annihilates ( Gamma1) k n k 3(k p) n : On this form, the identity can also be obtained directly as an application of the classical method of the previous section. Also note that the contiguity relation (6) can easily be obtained by the variant of Takayama s algorithm described in [4]. 2.3. The New Paradigm. In the case t = 3 it was noted that Zeilberger s algorithm fails to obtain the first order operator S n 3 for the sum (5) However, it is recovered once one notices n X k=0 f n;k = n X k=0 ( Gamma1) k n k 3k n (3k Gamma n(n 2) 0; 7) for ....

Chyzak, F., and Salvy, B. Non-commutative elimination in Ore algebras proves multivariate identities. J. Symbolic Comput. 26, 2 (1998), 187--227.

.... sequences, with counterparts for (possibly multiple) integrals and their q analogues [25, 26] As an example of application, Zeilberger s algorithm computes the following sum in closed form 2n X k=0 ( Gamma1) k 2n k 2k k 4n Gamma 2k 2n Gamma k = 2n n 2 : In [12], we described unified but rather slow algorithms based on skew Grobner basis calculations to perform creative telescoping in general classes of functions and sequences, including the class of holonomic functions. This can be viewed as a generalization of Zeilberger s slow algorithm. Our main ....

....telescoping computes a linear recurrence satisfied by the definite sum U n . In this article, we generalize Zeilberger s algorithm to the case when the linear equations satisfied by (u n;k ) have orders larger than 1, and are not necessarily recurrences. The definition of finite functions [12] is recalled in the next section. Next, we extend Abramov s alternative approach to Gosper s algorithm, then Zeilberger s algorithm to finite functions. We then detail how the normal forms for finite functions used in those algorithms are obtained by methods of Grobner bases. We finally ....

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Chyzak, F., and Salvy, B. Non-commutative elimination in Ore algebras proves multivariate holonomic identities. To appear. Preliminary version available as INRIA Research Report #2799, ftp://ftp.inria.fr/INRIA/publication/publi-ps-gz/RR/RR-2799.ps.gz.

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F. Chyzak, B. Salvy (1998). Non-commutative Elimination in Ore Algebras Proves Multivariate Identities. J. Symb. Comput. 26, no. 2, 187-227.

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F. Chyzak and B. Salvy. Non-commutative elimination in ore algebras proves multivariate identities. Journal of Symbolic Computation, 26(2):187--227, 1998.

No context found.

Frederic Chyzak and Bruno Salvy. Non-commutative elimination in Ore algebras proves multivariate holonomic identities. Journal of Symbolic Computation, 26(2):187--227, August 1998.

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F. Chyzak and B. Salvy. Non-commutative elimination in Ore algebras proves multivariate holonomic identities. Journal of Symbolic Computation, 26(2):187--227, 1998.

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Math. Phys. 105, 221-238. Chyzak, F. & Salvy, B. (1998). `Non-commutative elimination in Ore Algebras proves multivariate identities', J. Symbol.

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