B. Salvy and P. Zimmermann. Gfun: A Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software, 20(2):163--177, June 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....equation of order 4 2 which is identical to Eq. 86) Notice that by linear algebra one can deduce the difference equation of the product (90) given the difference equations of the factors, since they have polynomial coefficients. This can be done, e.g. by Maple command rec rec [35] of the gfun package. We conclude the proof by noticing that the results of the previous theorem can be used to extend Theorem 5 to the generalized co recursive associated of classical discrete orthogonal polynomials with real order of association as was done for classical continuous in Ref. 9] ....

B. Salvy and P. Zimmerman, GFUN: A Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software, 20 (1994), 163 -- 177.

.... the series Z 3 (t) is characterized by Z 3 (0) 1, Z t(1 Gamma t) 1 Gamma 9t) Z 3 (t) 0: 9 The series Z 4 (t) is characterized by Z 4 (0) 1, 4 (0) 4, 4 (t) 3t(1 Gamma 30t 128t These differential equations can be guessed using the gfun package of Maple [30]. In order to prove them, or equivalently, to prove the linear recurrence relations satisfied by their coefficients, it is convenient to use Bessel generating functions [11] Let z n denote the coefficient of t Z d (t) Then the Bessel generating function Y d (t) for these coefficients is Y ....

B. Salvy and P. Zimmermann, GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software 20 (1994) 163--167.

.... Y ) seems only natural if one suspects the expansion (5) However, we can alternatively obtain this expansion, without guessing it, via the following procedure. We convert the algebraic equation satised by Z(x; 1) into a linear dioeerential equation (using for instance the Maple package Gfun [18]) 27x Gamma 4) x; 1) 2x(27x Gamma 5) Z (x; 1) 2(3x Gamma 1)Z(x; 1) 2(3x Gamma 1) 0: Denoting by an the coeOEcient of x in Z(x; 1) this dioeerential equation gives: a 0 = 1; a 1 = 1; and for n 2; an = 3(3n Gamma 1) 3n Gamma 2) 2(n 1) 2n 1) an Gamma1 : 7) We ....

B. Salvy and P. Zimmermann. GFUN: a MAPLE package for the manipulation of generating and holonomic functions in one variable, ACM Transactions on Mathematical Software 20 (1994) 163167 (see also http://pauillac.inria.fr/algo/libraries/libraries.html).

....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 ....

B. Salvy and P. Zimmermann, Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software 20 (1994). (p. 3)

....problems. These matrix like approximation problems include Hermite Pad e, simultaneous Pad e and multi point Pad e approximation problems along with their vector and matrix generalizations [1, 4] Such computations appear in such diverse applications as the Gfun package of Salvy and Zimmerman [19] 21 for determining recurrences relations, factorization of linear differential operators [23] computation of matrix normal forms [24] inversion of structured matrices [16] and computation of common divisors of matrix greatest common divisors [7] ....

B. Salvy & P. Zimmermann, Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Transactions on Mathematical Software (TOMS), 20(2) (1994) 163-177.

.... 0 3 0 0 0 0 0 0 0 0 0 0 0 4 3 3 1 0 0 0 0 0 0 0 0 5 24 24 12 2 0 0 0 0 0 0 0 6 133 133 74 23 3 0 0 0 0 0 0 7 635 635 371 141 36 4 0 0 0 0 0 8 2807 2807 1688 709 227 51 5 0 0 0 0 9 11864 11864 7276 3248 1168 334 68 6 0 0 0 10 48756 48756 30340 14121 5459 1771 464 87 7 0 0 Using ordi of [10] or gfun[5] we conjecture that 123 (n) 2n(2n Gamma 1) n 5) n Gamma 4 It is very likely that one should be able to conjecture an explicit expression for P (n; I) which would be routine to prove, and from which the above conjecture would follow. 2 The Method We may now outline the method ....

.... 1; I Gamma 1) P (n Gamma 2; I) otherwise 0 0 1 0 0 2 0 0 0 3 1 1 0 0 4 5 5 2 0 0 5 21 21 11 3 0 0 6 84 84 49 19 4 0 0 7 330 330 204 92 29 5 0 0 8 1287 1287 825 405 153 41 6 0 0 9 5005 5005 3289 1705 715 235 55 7 0 0 10 19448 19448 13013 7007 3146 1166 341 71 8 0 0 Using ordi[10] or gfun[5] we conjecture that a 312 (n) the number of permutations on f1; 2; ng containing exactly one cab subsequence, Gamma 2n Gamma2 . 4 Counting abcd s Lest the reader think that the methods outlined in this paper will only help us gain information about permutations avoiding ....

B.Salvy and P. Zimmermann, Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Soft. 20 (1994).

....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 ....

B. Salvy and P. Zimmermann, Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software 20 (1994). (p. 3)

....operations making the algorithm suitable for implementation in computer algebra systems. This allows for efficient computation of matrix interpolation problems in the case of parameterized data. Such computations also appear in such diverse applications as the Gfun package of Salvy and Zimmerman [49] for determining recurrences relations, factorization of linear differential operators [54] and computation of matrix normal forms [55] The algorithm presented here is at least an order of magnitude faster than applying the fraction free algorithm of Bareiss [2] which is based on Gaussian ....

B. Salvy & P. Zimmermann, Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Transactions on Mathematical Software (TOMS), 20(2) (1994) 163-177.

....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 ....

....the holonomic function for which a HTML page is to be generated. Taylor expansion. It is classical that the Taylor coe#cients satisfy a linear recurrence that can be derived from the di#erential equation. This provides a fast method for Taylor expansion. This is implemented in the gfun package [8]. Closed form for Taylor coe#cients. From the recurrence above, Petkovsek s algorithm [6] decides whether the Taylor coe#cients u n are hypergeometric (i.e. u n 1 u n is a rational function of n) This is implemented in the LREtools package of Maple. Success then leads to a closed form for ....

Bruno Salvy and Paul Zimmermann. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software, 20(2):163--177, 1994.

.... computer algebra package MAGMA [6] d 3 5 7 9 11 13 15 17 19 21 23 25 A 6 (d) 1 3 5 9 15 25 41 67 109 177 287 465 A oe fl (d) 1 1 3 7 13 23 45 87 167 321 619 1193 A 3oe 4 (d) 1 1 5 7 21 37 89 173 383 777 1665 3441 Table 1 When we entered the values from the table into the Maple package gfun [23], we were able to guess a linear recurrence for the numbers A k (d) in each of the three cases. These recurrences are established by rather elaborate combinatorial arguments in Theorems 4.3, 4.6 and 4.8. Consequently, we obtain explicit formulas for the 2 ranks of the difference GAUSS SUMS, ....

....on the degrees of the numerator and denominator polynomials of the rational generating function P d2 R k (d)z d . So, this allows us, at least for k not too large, to determine recurrences via the computer, by first computing enough numbers R k (d) to guess a linear recurrence using gfun [23], and then applying Lemma 4.5. In the next theorem, we determine R k (d) explicitly for 3 k 9 by computing the corresponding rational generating functions. Theorem 7.5. For 3 k 9, the generating functions P d2 R k (d)z d are respectively X d2 R 3 (d)z d = z 2 (2 Gamma z) 1 ....

B. Salvy and P. Zimmermann, Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software 20 (1994).

....and presents it in a uniform way, with the added possibility of interactivity. We concentrate on special functions that are solutions of a linear differential or difference 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 ....

....holonomic function for which a HTML page is to be generated. Taylor expansion. It is classical that the Taylor coefficients satisfy a linear recurrence that can be derived from the differential equation. This provides a fast method for Taylor expansion. This is implemented in the gfun package [8]. Closed form for Taylor coefficients. From the recurrence above, Petkovsek s algorithm [6] decides whether the Taylor coefficients un are hypergeometric (i.e. un 1=un is a rational function of n) This is implemented in the LREtools package of Maple. Success then leads to a closed form for ....

Bruno Salvy and Paul Zimmermann. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software, 20(2):163--177, 1994.

....This in turn translates into a linear recurrence with polynomial coecients in n for the quantities [z n ]f . Thus, the coecient of index n of any algebraic function is computable in a number of operations that is linear in n. The procedure is implemented in Salvy and Zimmermann s Gfun package [67]. This remark applies to all the generating functions considered in this paper. For instance, the excursion generating function E(z) corresponding to the set of jumps f2; 1; 0; 1; 2g (Example 4) satis es an inhomogeneous di erential equation of order 3 (33) z 3 (5z 4) 5z 1) z 1) 2 (5z ....

Bruno Salvy and Paul Zimmermann, GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Transactions on Mathematical Software 20 (1994), no. 2, 163-167.

....for numerical and not exact computations, and so we have, on each example, to seek for numerical stability. This point uses a backward scheme which is developped on an example below. The computation of the coecients d n can be done eciently by nding a recurrence for example using the gfun package [9], because it is a composition of a known algebraic function and a function y known by its di erential equation. The initial conditions for the d n derive directly from the initial conditions of the di erential equation satis ed by y and so from the initial conditions of the di erential equation ....

Salvy (Bruno) and Zimmermann (Paul). { Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software, vol. 20, n 2, 1994, pp. 163{ 177.

....This in turn translates into a linear recurrence with polynomial coecients in n for the quantities [z n ]f . Thus, the coecient of index n of any algebraic function is computable in a number of operations that is linear in n. The procedure is implemented in Salvy and Zimmermann s Gfun package [67]. This remark applies to all the generating functions considered in this paper. For instance, the excursion generating function E(z) corresponding to the set of steps f2; 1; 0; 1; 2g satis es an inhomogeneous di erential equation of order 3 (33) z 3 (5z 4) 5z 1) z 1) 2 (5z 1) 2 d 3 E ....

Bruno Salvy and Paul Zimmermann, GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Transactions on Mathematical Software 20 (1994), no. 2, 163-167.

....of the function) is necessary when solving many computer algebra problems. We list some of these problems. P1. Expanding a function as a power series and subsequently investigating the expansion. An annihilator lets one construct the recurrence for the series coefficients and manipulate them ([14, 17]) P2. Solving linear inhomogeneous equations. Some methods use annihilators of the right hand side ( 4, 8] P3. Integrating. If the minimal annihilator L; ord L = n, of f is given, then one can check whether there exists a primitive of f with an n th order minimal annihilator. If yes, then it ....

B. Salvy, P. Zimmermann (1992): GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable, INRIA, Rapports Techniques, No 143.

....13, 23] 2. Let c n denote the coeOEcient of x n in C 0 (x) The numbers c n have large prime factors (see Table 1) We can prove they are not hypergeometric as follows: we rst construct the linear recurrence with polynomial coeOEcients they satisfy (using, for instance, the Maple package Gfun [18]) and then look for all hypergeometric solutions 15 of this recurrence (using the algorithm Hyper [15] We nd that there is no such solution: the sequence (c n ) n is not hypergeometric. This does not rule out the existence of an expression of the form c n = X k F n;k where F n;k would be ....

B. Salvy and P. Zimmermann, Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Transactions on Mathematical Software, 20 (1994) 163167.

.... Y ) 3 seems only natural if one suspects the expansion (5) However, we can alternatively obtain this expansion, without guessing it, via the following procedure. We convert the algebraic equation satised by Z(x; 1) into a linear dioeerential equation (using for instance the Maple package Gfun [18]) x 2 (27x Gamma 4) 2 Z x 2 (x; 1) 2x(27x Gamma 5) Z x (x; 1) 2(3x Gamma 1)Z(x; 1) 2(3x Gamma 1) 0: Denoting by an the coeOEcient of x n in Z(x; 1) this dioeerential equation gives: a 0 = 1; a 1 = 1; and for n 2; an = 3(3n Gamma 1) 3n Gamma 2) 2(n 1) 2n 1) ....

B. Salvy and P. Zimmermann. GFUN: a MAPLE package for the manipulation of generating and holonomic functions in one variable, ACM Transactions on Mathematical Software 20 (1994) 163167 (see also http://pauillac.inria.fr/algo/libraries/libraries.html).

.... robustness (as defined above) and the density of the graph 2 (i.e. the number of its edges) The originality of our approach consists in introducing in this range of problems methods of analytic combinatorics [14, 21] and recent research in automatic analysis based on symbolic computation [8, 13, 15, 23]. Additional threshold estimates regarding properties of multiple sourcedestination pairs are discussed in the last section of the paper. Summary of results. From earlier known results [7, 20] and this paper, a picture of robustness under the G n;p model emerges. As is usual in random graph ....

....with coefficients that are rational (equivalently polynomial) functions. Holonomic functions enjoy a rich set of closure properties, including closure under sums and products, integration and differentiation, as well as algebraic substitutions. The Maple package Gfun due to Salvy and Zimmermann [23] actually implements these closure properties. Here, since the exponential integral (also, its hypergeometric cognate) is clearly holonomic, we may take advantage of the Gfun package and build up automatically a differential equation satisfied by Q(z) z 4 z 5 4z 3 Gamma 1 Gamma z 4z ....

B. Salvy and P. Zimmermann, "GFUN: a Maple Package for the Manipulation of Generating and Holonomic Functions in One Variable", ACM Transactions on Mathematical Software 20(2), pp. 163--167, 1994. 20

....called gfun available in the Maple share library, which does all sorts of useful manipulations with sequences of numbers and the difference, differential, or algebraic equations for their generating functions. This package, written by Bruno Salvy and Paul Zimmermann from INRIA, described in [9], is particularly useful in experimental work. From the knowledge of an initial segment of a sequence of numbers only, and under the hypothesis of holonomicity, it may provide information in the form of intelligent guesses, and thus give the right ideas for further work. As an example, let us ask ....

Bruno Salvy and Paul Zimmermann, Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, Technical Report no. 143, INRIARocquencourt, 1992.

.... 0 3 0 0 0 0 0 0 0 0 0 0 0 4 3 3 1 0 0 0 0 0 0 0 0 5 24 24 12 2 0 0 0 0 0 0 0 6 133 133 74 23 3 0 0 0 0 0 0 7 635 635 371 141 36 4 0 0 0 0 0 8 2807 2807 1688 709 227 51 5 0 0 0 0 9 11864 11864 7276 3248 1168 334 68 6 0 0 0 10 48756 48756 30340 14121 5459 1771 464 87 7 0 0 Using ordi of [10] or gfun[5] we conjecture that a (2) 123 (n) 59n 2 117n 100 2n(2n Gamma 1) n 5) 2n n Gamma 4 : It is very likely that one should be able to conjecture an explicit expression for P (2) n; I) which would be routine to prove, and from which the above conjecture would follow. 2 The ....

.... Values of P (1) n; I) n I=0 1 2 3 4 5 6 7 8 9 10 0 0 1 0 0 2 0 0 0 3 1 1 0 0 4 5 5 2 0 0 5 21 21 11 3 0 0 6 84 84 49 19 4 0 0 7 330 330 204 92 29 5 0 0 8 1287 1287 825 405 153 41 6 0 0 9 5005 5005 3289 1705 715 235 55 7 0 0 10 19448 19448 13013 7007 3146 1166 341 71 8 0 0 Using ordi[10] or gfun[5] we conjecture that a (1) 312 (n) the number of permutations on f1; 2; ng containing exactly one cab subsequence, n Gamma2 2n Gamma 2n Gamma2 n Gamma1 Delta . 4 Counting abcd s Lest the reader think that the methods outlined in this paper will only help us gain information ....

B.Salvy and P. Zimmermann, Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Soft. 20 (1994).

....addition, multiplication, right composition with algebraic functions, di erentiation and integration, Hadamard product, etc. We refer to (Stanley, 1980; Lipshitz, 1989; Zeilberger, 1990) for more information on this subject. Holonomic functions are also available in some computer algebra systems (Salvy and Zimmermann, 1994). 3.4. Composition 3.4.1. Right composition with polynomials Let f = f 0 f p 1 z p 1 and g = g 1 z g q 1 z q 1 be polynomials, considered as truncated power series in TPS(C) In order to eciently compute f 0 n g 0 n : TPS(C; n) 16 Joris van der Hoeven for given n, we ....

Salvy, B., Zimmermann, P. (1994). Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Trans. on Math. Software, 20(2):163-177.

....exp # 1 m # e mx 1 # # exp # y # e mx 1 # # . We have thus verified equation (9) Acknowledgements. Without Sloane s integer sequence database I would probably never have come across the reference [Ri] Also at one instance the gfun Maple package by Salvy and Zimmermann [SZ] was helpful. ....

B. Salvy, P. Zimmermann, Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software 20 (1994), 163--177.

....(n Gamma k 2) D k n 2 (x) Gamma (2n 3) xD k n 1 (x) n k 1) D k n (x) 0 : 66) Currently this recurrence equation is not yet proved, but this will be done soon. Assume for the moment that E k n (x) are solutions of (66) Then by another application of linear algebra (see e.g. [20], 12] this recurrence equation can be squared , i.e. it is possible to calculate the recurrence equation of third order valid for E k n (x) 2 . This step can be accomplished, e.g. by the procedure rec rec of the gfun packacke ( 20] see also [10] with Maple, and results in the recurrence ....

....Then by another application of linear algebra (see e.g. 20] 12] this recurrence equation can be squared , i.e. it is possible to calculate the recurrence equation of third order valid for E k n (x) 2 . This step can be accomplished, e.g. by the procedure rec rec of the gfun packacke ([20], see also [10] with Maple, and results in the recurrence equation ( 2 n 5 ) k n 2 ) k n 1 ) 2 S k n (x) Gamma ( 2 n 3 ) k n 2 ) k 2 Gamma n 2 4 x 2 n 2 Gamma 4 n 16 x 2 n 15 x 2 Gamma 4 ) S k n 1 (x) Gamma ( 2 n 5 ) Gamma2 k Gamma n ) ....

Salvy, B. und Zimmermann, P.: GFUN: A Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software 20, 1994, 163--177.

....For instance, having a computer algebra system at hand, one could examine the integer factorizations of the ratios EO(m 1) EO(m) or EO(m 2) EO(m) etc. for some small values of m and then look out for a pattern. Here programs like the Maple procedure gfun written by Salvy and Zimmermann, see e.g. Salvy and Zimmermann (1993), might be helpful. Suppose our guess is, as above, 2 4m Gamma 4m 2m Delta Gamma1 then using Zeilberger s algorithm it is easy to check whether the recursion boils down or not. For instance: In[13] Zb[2( 4m) Binomial[4m,2m] 4 ( 1)j Binomial[m 1,j] Binomial[2m 1,2j] ....

Salvy, B., and Zimmermann, P. (1993). Gfun: A Maple package for the manipulation of generating and holonomic functions in one variable. to appear in: ACM Transactions on Mathematical Software.

.... series of about 1000 out of the 4568 sequences appearing in [Sloane] Since the first version of this article was written, a Maple package implementing some ideas presented here, as well as others such as the D finite approach, has been written by Bruno Salvy and Paul Zimmermann of INRIA [Salvy and Zimmermann] It is now available as a shared package under the name gfun . To learn more about obtaining shared packages, type share to Maple. The analogue of our function generating in gfun is the function guessgf. Giving guessgf the right set of options results in its using the set of transformations ....

B. Salvy and P. Zimmermann, "GFUN: A Maple Package for the Manipulation of Generating and Holonomic Functions in one Variable", to appear in ACM Trans. in Math. Software.

....the sum in the first line of (6.13) Carsten Schneider s extension of Karr s algorithm [19] implemented by Schneider, finds a recurrence for the double sum in the second line of (6. 13) Finally, Mallinger s Mathematica package GeneratingFunctions [29] or Salvy and Zimmermann s Maple package gfun [45] can be used to combine these two recurrences into one, a recurrence of order 10. It is then routine to check (preferably on the computer) that the right hand side of (6.13) satisfies this same recurrence. However, in the present implementation, these algorithms are not able to find the explicit ....

B. Salvy and P. Zimmermann, Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software 20 (1994).

.... Z 0 4 (0) 4, Z 00 4 (0) 56 and t 2 (1 Gamma 4t) 1 Gamma 16t) Z 000 4 (t) 3t(1 Gamma 30t 128t 2 ) Z 00 4 (t) 1 Gamma 68t 448t 2 ) Z 0 4 (t) Gamma 4(1 Gamma 16t) Z 4 (t) 0: These differential equations can be guessed using the gfun package of Maple [30]. In order to prove them, or equivalently, to prove the linear recurrence relations satisfied by their coefficients, it is convenient to use Bessel generating functions [11] Let z (d) n denote the coefficient of t n in Z d (t) Then the Bessel generating function Y d (t) for these ....

B. Salvy and P. Zimmermann, GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software 20 (1994) 163--167.

....j Gamma 2l i 2i Gamma j y i j 2i Gamma j = 0; An alternative evaluation of the Andrews Burge determinant 5 for 1 j n Gamma 1, 1 l j=2. This identity can be proved routinely by means of Zeilberger s algorithm [18, 19] and Salvy and Zimmermann s Maple package GFUN [17]. However, it happens that a 4 F 3 summation is already known that applies to both 4 F 3 series in (3.3) namely Lemma 1 in [9] Here we need the assumption l j=2. Little simplification then establishes (3.3) and hence the claim. Thus, Q n Gamma1 j=1 ( x y) 2 j 1) b(j 1) 2c is a ....

B. Salvy and P. Zimmermann, GFUN --- A MAPLE package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software 20 (1994), 163--177.

....under Hadamard (i.e. term wise) product and under diagonal. All these results are proved by Stanley in [22] P recursive sequences also form an algebra and possess corresponding properties. These interesting properties have led Salvy and Zimmermann to implement the Gfun Maple package described in [21]. This package manipulates sequences, linear recurrence equations or linear differential equations and generating functions of various types. In particular, they have implemented algorithms that compute the sum, product and Hadamard product of holonomic functions in a single variable and sum, ....

Salvy, B., and Zimmermann, P. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. Technical Report 143, Institut National de Recherche en Informatique et en Automatique, 1992. To appear in ACM Transactions on Mathematical Software.

....Y ) 3 seems only natural if one suspects the expansion (5) However, we can alternatively obtain this expansion, without guessing it, via the following procedure. We convert the algebraic equation satisfied by Z(x, 1) into a linear di#erential equation (using for instance the Maple package Gfun [18]) x 2 (27x 4) # 2 Z #x 2 (x, 1) 2x(27x 5) #Z #x (x, 1) 2(3x 1)Z(x, 1) 2(3x 1) 0. Denoting by an the coe#cient of x n in Z(x, 1) this di#erential equation gives: a 0 =1,a 1 =1, and for n # 2,a n = 3(3n 1) 3n 2) 2(n 1) 2n 1) an 1 . 7) We then ....

B. Salvy and P. Zimmermann. GFUN: a MAPLE package for the manipulation of generating and holonomic functions in one variable, ACM Transactions on Mathematical Software 20 (1994) 163--167 (see also http://pauillac.inria.fr/algo/libraries/libraries.html).

....equations that we had. The last step, that of extracting coe#cients in exact form, was, at large, the least systematic and mechanical one. A great deal of combinatorial identities, inspired guessing and patience was needed. Standard Maple tools like the function interp or the Gfun package [29] proved also to be useful. However, once the solution is obtained, it is just a matter of minutes to check its correctness. It is quite di#cult to provide a detailed and ordered description of the methods that we used to extract coe#cients from generating functions. As a result, the paper contains ....

B. Salvy and P. Zimmermann. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software, 20(2):163--177, 1994.

....number field, then y is also defined by p(x; y) 0 for some polynomial p with rational coefficients. It should be pointed out that the coefficients of an algebraic series are indeed algebraic numbers, and that the coefficients satisfy a recurrence. The following is noted in [2] and see also [7]. If f is an algebraic series there exists a positive integer n 0 and polynomials p 0 ; p d with coefficients in Q such that for all integers n n 0 , p 0 (n) Delta [x n ]f p 1 (n) Delta [x n Gamma1 ]f Delta Delta Delta p d (n) Delta [x n Gammad ]f = 0 : The question of ....

Bruno Salvy and Paul Zimmermann. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software, 20(2):163--177, 1994.

....Matrix Pad e Approximants. We should point out that computing rational approximants and interpolants along with their vector and matrix generalizations has many applications in computer algebra computations. Such computations appear in such applications as the Gfun package of Salvy and Zimmerman [46] for determining recurrences relations, computation of matrix greatest common divisors [9] and the factorization of linear differential operators having rational function coefficients [50] In all cases there is a general theme that runs through our search for efficient algorithms for such linear ....

B. Salvy & P. Zimmermann, Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Transactions on Mathematical Software (TOMS), 20(2) (1994) 163-177.

....Y ) 3 seems only natural if one suspects the expansion (5) However, we can alternatively obtain this expansion, without guessing it, via the following procedure. We convert the algebraic equation satisfied by Z(x; 1) into a linear differential equation (using for instance the Maple package Gfun [18]) x 2 (27x Gamma 4) 2 Z x 2 (x; 1) 2x(27x Gamma 5) Z x (x; 1) 2(3x Gamma 1)Z(x; 1) 2(3x Gamma 1) 0: Denoting by an the coefficient of x n in Z(x; 1) this differential equation gives: a 0 = 1; a 1 = 1; and for n 2; an = 3(3n Gamma 1) 3n Gamma 2) 2(n 1) 2n 1) ....

B. Salvy and P. Zimmermann. GFUN: a MAPLE package for the manipulation of generating and holonomic functions in one variable, ACM Transactions on Mathematical Software 20 (1994) 163--167 (see also http://pauillac.inria.fr/algo/libraries/libraries.html).

....More details will be given in the following subsections. 2.3.2. Computing S(t) From the equation S = A(t; S) it is easy to compute the first L coefficients of S as a formal series in t, by an iterative process in O(L 2 ) operations. We can do better using standard techniques from combinatorics [11, 46]. In particular, S satisfies a second order linear differential equation with polynomial coefficients in t, from which we can easily deduce recurrence relations between the coefficients of S. Hence, these coefficients can be computed in O(L) operations modulo precomputations. See section 10.1 for ....

....if i = 1; 2; 3; 5; 1 a 6 if i = 4; S 2i a 6 i S 2i Gamma5 S 2 i Gamma1 P i Gamma2 j=4 S 2 j S 2i Gamma2j 1 j otherwise; 27) and S 2i = 1 if i = 1; 2; 3; 4; 5; S 2i Gamma1 a 6 i S 2i Gamma6 P i Gamma2 j=4 S 2 j S 2i Gamma2j j otherwise. 28) Using standard tools [46], we also find that S(t) satisfies the diffential equation: Gamma Gamma54 t 5 Gamma 4 t 3 14 t 2 Gamma 18 t 8 Delta y Gamma Gamma48 t 54 t 2 Gamma 28 t 3 6 t 4 54 t 6 16 Delta y 0 Gamma Gamma27 t 8 28 t 2 Gamma 36 t 3 20 t 4 Gamma ....

Salvy, B., and Zimmermann, P. Gfun: a maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software 20, 2 (1994), 163--177.

....equations that we had. The last step, that of extracting coefficients in exact form, was, at large, the least systematic and mechanical one. A great deal of combinatorial identities, inspired guessing and patience was needed. Standard Maple tools like the function interp or the Gfun package [27] proved also to be useful. However, once the solution is obtained, it is just a matter of minutes to check its correctness. It is quite difficult to provide a detailed and ordered description of the methods that we used to extract coefficients from generating functions. As a result, the paper ....

B. Salvy and P. Zimmermann. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software, 20(2):163-- 177, 1994.

....guessing a recurrence on the basis of the first few values of the sequence and similar tasks, there is a package called gfun of procedures available in the Maple share library, accessible via anonymous ftp at daisy.uwaterloo.ca or neptune.inf.ethz.ch. It is due to B. Salvy and P. Zimmerman [44] and contains a nice set of tools for the combinatorialist working with sequences of numbers and their recurrences, the corresponding generating functions and the differential equations they satisfy. 2 Binomial Identities 2.1 Generalities Binomial summations, or combinatorial sums , their ....

B. Salvy and P. Zimmerman. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. preprint, October 1992.

.... 3 4x 4 8x 5 which leads to P (x) 1 Gamma 2x x 2 Gamma x 4 Gamma x 2 p 1 Gamma 4x 2 (1 x) 2 (1 Gamma 2x) 2 : Let us mention that the package GFUN in Maple is able to make such translations (recurrences, differential equations, algebraic equations, closed forms) see [23]. 4. Temperley s Method We are going to illustrate Temperley s method [24] with the enumeration of column convex polyominoes (on a square lattice) with respect to perimeter [7] The generating function G(y) X n2 y 2n can be rewritten as G(y) X r1 g r (y) where the g r satisfy a ....

Salvy (Bruno) and Zimmermann (Paul). -- Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software, vol. 20, n 2, 1994, pp. 163-- 177. -- ftp://ftp.inria.fr/INRIA/publication/publi-ps-gz/RT/RT-0143.ps.gz.

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B. Salvy and P. Zimmermann. Gfun: A Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software, 20(2):163--177, June 1994.

....linear recurrence Mn = 2n Gamma 1 n 1 Mn Gamma1 3n Gamma 6 n 1 Mn Gamma2 satisfied by the numbers Mn . Such a recurrence exists for any context free grammar (i.e. when only the union and product constructions are used) and it can be computed from the grammar using the Gfun package [16]. Thanks to the IEEE 754 standard, the computed lower and upper bounds are guaranteed to be exact, but differ according to the way of computation. The following table indicates for different sizes the accuracy 7 Gamma lg 4;n with 4;n as in Proposition 3.2 for the nonterminal T 4 = M , i.e. ....

.... 0:056 5000 NA=35: 82 66=0:08 0:163 10000 NA=162 586 282=0:18 0:411 Fit n 3:6 =n 2:3 n 3:2 n 2:0 =n 1:2 n 1:1 In the count column, the times on the left were obtained with the default O(n 2 ) method, and those on the right with the linear recurrence computed by the Gfun package [16], after typing combstruct usegfun : true in Maple. INRIA Uniform Random Generation of Decomposable Structures Using Floating Point Arithmetic 15 6 Conclusion and open questions In this paper, we have extended to certified floating point computations the recursive method for the random ....

Salvy, B., and Zimmermann, P. Gfun: A Maple package for the manipulation of generating and holonomic functions in one variable. ACM Trans. Math. Softw. 20, 2 (June 1994), 163--177.

....Q. 2. Rewrite this recurrence as a linear recurrence of order 1 relating the vector Un =#un ; u n,m 1#t o Un,1 by Un = AUn,1 where A is a constant m # m matrix. 3. Use binary powering to compute the power of A in Un = A n,m Um . This operation is implemented in the Maple package gfun #Salvy Zimmermann 1994#. As an example, Fig. 1 displays the probability that the pattern ACAGAC occurs exactly twice in a text over the alphabet fA,C,G,Tg against the length n of the text. The probabilities assigned to each of the letters are taken from a viral DNA ##X174#. The shape of the curveistypical of that ....

Salvy, B., and Zimmermann, P. 1994. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software 20#2#:163#177.

....questions turn up systematically in the reversion of power series since the inversion theorem of Lagrange states that the nth coefficient in the expansion of the inverse of a function f is expressible as an nth coefficient in an expression that involves an nth power of f . We use the gfun package [7] 1 that addresses the problem of manipulating series that satisfy linear differential equations with polynomial or rational coefficients (these are often called holonomic functions) From the point of view of symbolic manipulation, the importance of this package lies in the fact that: i) a ....

Salvy, B., and Zimmermann, P. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software 20, 2 (1994), 163--177.

....both the algorithm of Section 1 and the algorithm of Section 2 will soon be part of the Maple share library. Combinatorialists might use it fruitfully in conjunction with the gfun package which provides tools for manipulating linear recurrent sequences and linear differential equations [8]. Another useful application of this program is in conjunction with Zeilberger s technique of creative telescoping [11] Thus one could get a closed form solution to some definite hypergeometric summations. As a final note, our algorithms extend without any modification to fields of ....

Salvy, B., and Zimmermann, P. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. Technical Report 143, Institut National de Recherche en Informatique et en Automatique, 1992. To appear in ACM Transactions on Mathematical Software.

....as a linear recurrence of order 1 relating the vector Un = u n ; un Gammam 1 ) to Un Gamma1 by Un = AUn Gamma1 where A is a constant m Theta m matrix. 3. Use binary powering to compute the power of A in Un = A n Gammam Um . This operation is implemented in the Maple package gfun (Salvy Zimmermann 1994). As an example, Fig. 1 displays the probability that the pattern ACAGAC occurs exactly twice in a text over the alphabet fA,C,G,Tg against the length n of the text. The probabilities assigned to each of the letters are taken from a viral DNA (OEX174) The shape of the curve is typical of that ....

Salvy, B., and Zimmermann, P. 1994. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software 20(2):163--177.

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Salvy, B. and Zimmermann, P.: GFUN: A Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software 20 (1994), 163--177.

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B. Salvy and P. Zimmermann, Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software 20 (1994).

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Bruno Salvy and Paul Zimmermann, GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Transactions on Mathematical Software 20 (1994), no. 2, 163-167.

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Bruno Salvy and Paul Zimmermann. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software, 20(2):163-177, 1994.

No context found.

Salvy, B., Zimmermann, P. (1994). GFUN: A Maple Package for the Manipulation of Generating and Holonomic Functions in One Variable. Trans. Math. Soft. 20, No. 2, 163-177.

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Salvy, B., and Zimmermann, P. Gfun: a maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software 20, 2 (1994), 163--177.

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