F. Chyzak, Holonomic systems and automatic proofs of identities, INRIA Research Report no. 2371, 61 pp, 1994. (p. 3)  Home/Search   Document Details and Download   Summary   Related Articles   Check

....see for example Appendix B. Since the value of this book may not be easy to appreciate because of heavy notation, we refer the reader to [102] for a clarification of the notation 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 ....

F. Chyzak, Holonomic systems and automatic proofs of identities, INRIA Research Report no. 2371, 61 pp, 1994. (p. 3)

....Since the value of this book may not be easy to appreciate because of heavy notation, we refer the reader to [102] for a clarification of the notation and a clear presentation of many 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 ....

F. Chyzak, Holonomic systems and automatic proofs of identities, INRIA Research Report no. 2371, 61 pp, 1994. (p. 3)

....results in this direction are the calculation of the derivative rules for the Jacobi polynomials P (ff;fi) n (x) see Theorem 8, since this is a 6 variable problem. Note that, in a similar treatment, Chyzak tried to derive this type of result with a Maple implementation, without success ([5], x4.1) Using the Reduce implementation [21] any of these calculations (for the Jacobi polynomials) needs about five minutes on a DEC Alpha workstation (using the ezgcd switch, see [10] x9.3) Note that also with a pure lexicographic term order, the Grobner bases are derived in a similar time. ....

Chyzak, F.: Holonomic systems and automatic proofs of identities. Rapport de Recherche 2371, INRIA Research Report, Rocquencourt, 1994, available via anonymous ftp on ftp.inria.fr.

....of the lack of the term F (n 1) still be solved by iteration. It turns out that four cases (n; j even or odd) have to be distinguished. The third entry is a double sum and thus not so easily treatable. I recently showed it to F. Chyzak, and he is able to handle it with his package mgfun (see [1]) However, because of the following argument, the double sum is reducible to the other instances. We will show that the three statistics are basically the same. Assume that there are i nodes with 0 successors, j nodes with 1 successor, and k nodes with 2 successors. Then we have by an elementary ....

F. Chyzak. Holonomic systems and automatic proofs of identities. Research Report 2371, Institut National de Recherche en Informatique et en Automatique, October 1994.

.... 2 F[x] 4 x F [x] F [x] 0 In[7] HolonomicDE[Sin[x y] Sin[x]Sin[y] Cos[x]Cos[y] x] 3) Out[7] 4 F [x] F [x] 0 In[8] HolonomicDE[Cos[x y] Sin[x]Cos[y] Cos[x]Sin[y] x] 3) Out[8] 4 F [x] F [x] 0 In[9] HolonomicDE[Cos[y] Sin[y] y] 3) Out[9] 4 F [y] F [y] 0 In[10]: HolonomicDE[Sin[2y] 2,y] Out[10] 4 F[y] F [y] 0 One difficulty that may arise with the method described is that in some instances the sum and product algorithms will not generate the holonomic differential equation of lowest order, as in the above example for cos y sin y. In this case, ....

.... 0 In[7] HolonomicDE[Sin[x y] Sin[x]Sin[y] Cos[x]Cos[y] x] 3) Out[7] 4 F [x] F [x] 0 In[8] HolonomicDE[Cos[x y] Sin[x]Cos[y] Cos[x]Sin[y] x] 3) Out[8] 4 F [x] F [x] 0 In[9] HolonomicDE[Cos[y] Sin[y] y] 3) Out[9] 4 F [y] F [y] 0 In[10] HolonomicDE[Sin[2y] 2,y] Out[10]= 4 F[y] F [y] 0 One difficulty that may arise with the method described is that in some instances the sum and product algorithms will not generate the holonomic differential equation of lowest order, as in the above example for cos y sin y. In this case, the normal form property is lost. In ....

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Chyzak, F.: Holonomic systems and automatic proofs of identities. Rapport de Recherche 2371, INRIA Research Report, Rocquencourt, available via anonymous ftp on ftp.inria.fr (1994).

....(An instance of a holonomic system are the Gegenbauer polynomials C ff n (x) given in Example 1.3.1) Since this thesis concentrates on holonomic univariate functions, we omit exact definitions of multivariate holonomicity. Detailed descriptions and discussions were given, for example, by Chyzak [Chy94], Gessel [Ges90] Lipschitz [Lip89] and Zeilberger [Zei90] Chapter 2 My Mathematica Package 2.1 Introduction In Chapter 1 we discussed some properties of holonomic sequences and their generating functions. A closer look at the proofs of these properties reveals the fact that these proofs ....

F. Chyzak. Holonomic systems and automatic proofs of identities. Rapport de recherche 2371, INRIA, Rocquencourt, 1994.

....with respect to the argument (Zeilberger 1990) One is then led to consider systems of linear operators and algebras of such operators. Under mild conditions, the left ideals of these algebras are finitely generated and a non commutative variant of Buchberger s algorithm works in this context (Chyzak 1994). Many of the closure properties of the univariate case still hold and some of them have been implemented by Chyzak in the Mgfun package. Identities satisfied by combinatorial sequences like Ap ery s sequence (see Van der Poorten (1979) can be obtained almost routinely by elimination using ....

Chyzak, F. (1994, October). Holonomic systems and automatic proofs of identities. Research Report 2371, Institut National de Recherche en Informatique et en Automatique.

....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 algorithm that computes recurrences for double sums of hypergeometric functions in incredibly short time. 1.2 Notation and Basic Definitions In this ....

F. Chyzak. Holonomic systems and automatic proofs of identities. Research Report 2371, Institut National de Recherche en Informatique et en Automatique, October 1994.

....not guaranteed to succeed. We give two such algorithms. The first one is slow but will always terminate (successfully or detecting that the algorithm has failed) the second one is faster but may fail to terminate. All these operations are illustrated by examples using F. Chyzak s implementation [6] 1 . In conclusion, we recall the special case of Weyl algebras, where more operations are possible and we discuss envisioned extensions. 1. Non commutative algebras of operators 1.1. Definitions. O. Ore [20] initiated an algebraic treatment of a very general class of linear operators now ....

....algorithm. When this theorem applies, efficiency can be improved by suitable generalizations of the so called normal strategy [9, chap. 2] sugar strategy [12] and by trace lifting [30] Further discussion of implementation and efficiency will be part of F. Chyzak s thesis (see also [6]) Theorem 2. Let O = K(x) y] oe; ffi] be an Ore algebra over a field K such that , oe and ffi satisfy relations of the type i y j = a i;j y j i b i;j ; 1 i r; 1 j q; with a i;j 6= 0 in K(x) and b i;j in K(x) y] Then O is left Noetherian and a non commutative version of ....

Chyzak, F. Holonomic systems and automatic proofs of identities. Research Report 2371, Institut National de Recherche en Informatique et en Automatique, Oct. 1994.

.... at a general level, thereby setting the emphasis back on general holonomic and finite functions, as opposed to hypergeometric and q hypergeometric ones; iii) the extension and improvement of an algorithm by Takayama for fast creative telescoping; iv) a Maple implementation Mgfun, due to (Chyzak, 1994) y , which makes it easy to work in various mixed contexts with a single program. All the operations described in this paper are illustrated by examples using this package. The present article is organized as follows. In Section 1, Ore polynomials are introduced and the algorithmic tools to work ....

....efficiency can be improved by suitable generalizations of the so called normal strategy (Cox et al. 1992, Chap. 2) sugar strategy (Giovini et al. 1991) and by trace lifting (Takayama, 1995) Further discussion of implementation and efficiency will be part of (Chyzak, 1998) see also (Chyzak, 1994). As can be seen from Table 2, this theorem applies to many useful Ore algebras. The only exception mentioned in this table is the algebra K[x] M ; M; 0] for a Mahlerian operator M . Compare to K(x) M ; M; 0] which is Euclidean and hence Noetherian. The special case p = 0 of this theorem states ....

Chyzak, F. (1994). Holonomic systems and automatic proofs of identities. Research Report 2371, Institut National de Recherche en Informatique et en Automatique.

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