The Gfun Package

Presentation

Gfun is a Maple package that provides tools for

guessing a sequence or a series from its first terms;
manipulating rigorously solutions of linear differential or recurrence equations, using the equation as a data-structure.

Download gfun 3.62 (January 2012)

The latest version of gfun can be downloaded from here. It consists of two files:

Once downloaded, see the help page of libname to understand how to make them available from Maple. Typically, your session will contain something like libname:="gfun_path",libname: You can check that this worked by asking gfun:-version(); This should return the number above.

Reference for Gfun

The primary reference to use when citing gfun is the following one.

1994

@article{SaZi94,
    author      = {Salvy, Bruno and Zimmermann, Paul},
    title       = {Gfun: a {M}aple package for the manipulation of generating
  and holonomic functions in one variable},
    journal     = {ACM Transactions on Mathematical Software},
    number      = {2},
    year        = {1994},
    volume      = {20},
    pages       = {163--177},
    doi         = {10.1145/178365.178368},
    abstract    = {We describe the GFUN package which contains functions for
  manipulating sequences, linear recurrences, or differential equations and
  generating functions of various types. This article is intended both as an
  elementary introduction to the subject and as a reference manual for the
  package.},
}

BibTeX

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 n°2:163–177, 1994. — Reprint. (doi:10.1145/178365.178368)

We describe the GFUN package which contains functions for manipulating sequences, linear recurrences, or differential equations and generating functions of various types. This article is intended both as an elementary introduction to the subject and as a reference manual for the package.

If you are using the NumGfun subpackage, then the proper reference is:

2010

@inproceedings{Mezzarobba2010,
    author      = {Mezzarobba, Marc},
    title       = {Num{G}fun: a Package for Numerical and Analytic Computation
  with {D}-finite Functions},
    booktitle   = {Proceedings of the 2010 International Symposium on Symbolic
  and Algebraic Computation (ISSAC 2010)},
    year        = {2010},
    publisher   = {ACM},
    institution = {arXiv},
    pages       = {139--145},
    doi         = {10.1145/1837934.1837965},
    arxiv       = {abs/1002.3077},
    abstract    = {This article describes the implementation in the software
  package NumGfun of classical algorithms that operate on solutions of linear
  differential equations or recurrence relations with polynomial coefficients,
  including what seems to be the first general implementation of the fast
  high-precision numerical evaluation algorithms of Chudnovsky and Chudnovsky.
  In some cases, our descriptions contain small improvements over existing
  algorithms. We also provide references to relevant ideas not currently used
  in NumGfun.},
}

BibTeX

Marc Mezzarobba

. — NumGfun: a package for numerical and analytic computation with D-finite functions. — In Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation (ISSAC 2010), pages 139–145. ACM, 2010. (arXiv:abs/1002.3077) (doi:10.1145/1837934.1837965)

This article describes the implementation in the software package NumGfun of classical algorithms that operate on solutions of linear differential equations or recurrence relations with polynomial coefficients, including what seems to be the first general implementation of the fast high-precision numerical evaluation algorithms of Chudnovsky and Chudnovsky. In some cases, our descriptions contain small improvements over existing algorithms. We also provide references to relevant ideas not currently used in NumGfun.

Articles citing Gfun

The following is a list of articles that cite gfun. If you know of an article that is not in this list, but does cite gfun, please send us an email so that we add it to this list.

2011

Bernardi (O.), Bousquet-Mélou (M.) - Counting colored planar maps: Algebraicity results. Journal of Combinatorial Theory, Series B, vol. 101, n°5, pages 315–377.
Hebisch (W.), Rubey (M.) - Extended Rate, more Gfun. Journal of Symbolic Computation, vol. 46, n°8, pages 889–903.
Kauers (M.), Paule (P.) - The Concrete Tetrahedron. Series Texts and Monographs in Symbolic Computation, 203 pages.

2010

Bostan (A.), Kauers (M.), van Hoeij (M.) - The complete generating function for Gessel walks is algebraic. Proceedings of the American Mathematical Society, vol. 138, n°9, pages 3063–3078.
Denise (A.), Ponty (Y.), Termier (M.) - Controlled non-uniform random generation of decomposable structures. Theoretical Computer Science, vol. 411, n°40-42, pages 3527–3552.
Mezzarobba (M.) - NumGfun: A package for numerical and analytic computation with D-finite functions. International Symposium on Symbolic and Algebraic Computation, ISSAC 2010, Munich, July, pages 139–146.

2009

Blümlein (J.), Kauers (M.), Klein (S.), Schneider (C.) - Determining the closed forms of the $O(a_s^3)$ anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra. Computer Physics Communications, vol. 180, n°11, pages 2143–2165.
Bostan (A.) and Kauers (M.) - Automatic classification of restricted lattice walks. DMTCS Proceedings of the 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC'09), pages 203–217.
Gerhold (S.), Warnung (R.) - Finding efficient recursions for risk aggregation by computer algebra. Journal of Computational and Applied Mathematics, vol. 223, n°1, pages 499–507.
Kauers (M.), Koutschan (C.) - A Mathematica package for q-holonomic sequences and power series. Ramanujan Journal, vol. 19, n°2, pages 137–150.
Koseleff (P.-V.), Pecker, D. - A polynomial parametrization of torus knots. Applicable Algebra in Engineering, Communications and Computing, vol. 20, n°5-6, pages 361–377.
Mishna (M.). - Classifying lattice walks restricted to the quarter plane. Journal of Combinatorial Theory, Series A, vol. 116, n°2, pages 460–477.
Rechnitzer, (A.) - The anisotropic generating function of self-avoiding polygons is not D-finite. Polygons, Polyominoes and Polycubes, A. J. Guttman (ed.), vol. 775 of Lecture Notes in Physics, pages 93–115.
van der Hoeven (J.). - On asymptotic extrapolation. Journal of Symbolic Computation, vol. 44, n°8, pages 1000–1016.

2008

Ayyer (A.) and Zeilberger (D.) - Two dimensional directed lattice walks with boundaries. Tapas in Experimental Matehmatics, Contemporary Mathematics Series, vol. 457, pages 1–19.
Bernardi (O.) - On Triangulations with High Vertex Degree. Annals of Combinatorics, vol. 12, n°1, pages 17–44.
Borwein (J. M.) and Salvy (B.). - A proof of a recursion for Bessel moments. Experimental Mathematics, vol. 17, n°2, 223–230.
Carette (J.), Smith (S.), McCutchan (J.), Anand (C.), Korobkine (A.) - Case studies in model manipulation for scientific computing. Intelligent Computer Mathematics, vol. 5144 of Lecture Notes in Computer Science, pages 24–37.
Egorychev (G. P.) and Zima (E. V.). - Integral representation and algorithms for closed form summation. Handbook of Algebra, vol. 5, North-Holland, pages 459–529.
Gerhold (S.), Glebsky (L.), Schneider (C.), Weiss (H.), Zimmermann (B.) - Computing the complexity for Schelling segregation models. Communications in Nonlinear Science and Numerical Simulation, vol. 13, n°10, pages 2236–2245.
Kauers (M.) and Zimmermann (B.) - Computing the algebraic relations of C-finite sequences and multisequences. Journal of Symbolic Computation, vol 43, pages 787–803.
Lorenz (W. A.), Ponty (Y.), Clote (P.) - Asymptotics of RNA shapes. Journal of Computational Biology, vol. 15, n°1, pages 31–63.
Ponty (Y.). - Efficient sampling of RNA secondary structures from the Boltzmann ensemble of low-energy: the boustrophedon method. Journal of Mathematical Biology, vol. 56, 1–2, pages 107–127.
Prodinger (H.). - On the order of the recursion relation of Motzkin numbers of higher rank. Open Applied Mathematics Journal, vol. 2, pages 14–15.
Rao (N.) and Edelman (A.). - The polynomial method for random matrices. Foundations of Computational Mathematics vol. 8, n°6, pages 649–702.
Zhang (J.). - Sumudu transform based coefficients calculation. Nonlinear Studies vol. 15, n°4, pages 355–372.

2007

Bodirsky (M.), Gröpl (C.), Johannsen (D.), and Kang (M.). - A direct decomposition of 3-connected planar graphs. Séminaire Lotharingien de Combinatoire, vol. 54a, pages 1–15.
Carette (J.) and Janicki (R.) - Computing Properties of Numerical Imperative Programs by Symbolic Computation. Fundamenta Informaticae, vol. 80, n°1–3, pages 125–146.
Schneider (C.). - Symbolic summation assists combinatorics. Séminaire Lotharingien de Combinatoire, vol. 56, n°B56b, pages 1–36.
Fusy (É.) - Counting Unrooted Maps Using Tree-Decomposition. Séminaire Lotharingien de Combinatoire, vol. 54A, pages 1–44.
Kauers (M.). - An algorithm for deciding zero equivalence of nested polynomially recurrent sequences. ACM Transactions on Algorithms, vol. 3, 2, art. 1240241.
Kauers (M.). - Computer algebra and power series with positive coefficients. Formal Power Series and Algebraic Combinatorics (Tianjin, China), pages 1–7.
Koepf (W.). - Bieberbach's conjecture, the de Branges and Weinstein functions and the Askey-Gasper inequality. Ramanujan Journal, vol. 13, 1–3, pages 103–129.
Koepf (W.), Rajkovic (P. M.), Marinkovic (S.D.). Properties of q-holonomic functions. Journal of Difference Equations and Applications, vol. 13, n°7, pages 621–638.
Mansour (T.), Schork (M.) and Sun (Y.) - Motzkin numbers of higher rank: generating function and explicit expression. Journal of Integer Sequences vol. 10, n°7, Article 07.7.4, 11 pages.
Paule (P.) and Schneider (C.). - Truncating binomial series with symbolic summation. Electronic Journal of Combinatorial Number Theory vol. 7, n°A22, pages 1–9.
Schneider (C.). - Apéry's double sum is plain sailing indeed. Electronic Journal of Combinatorics, vol. 14, 1, pages 1–3.
Zeilberger (D.). - The Holonomic Ansatz II. Automatic Discovery(!) and Proof(!!) of Holonomic Determinant Evaluations. Annals of Combinatorics, vol. 11, n°2, pages 241–247.

2006

Becirovic (A.), Paule (P.), Pillwein (V.), Riese (A.), Schneider (C.), Schöberl (J.). Hypergeometric summation algorithms for high-order finite elements. Computing vol. 78, n°3, pages 235–249.
Bousquet-Mélou (M.). - Three osculating walkers. Journal of Physics: Conference Series, vol. 42, n°1, art. 5, pages 37–46.
Bousquet-Mélou (M.). - Rational and algebraic series in combinatorial enumeration. International Congress of Mathematicians, Invited paper, pages 789–826.
Bousquet-Mélou (M.). - Three osculating walkers. Journal of Physics: Conference Series, vol. 42, pages 35–46.
Bousquet-Mélou (M.) and Xin (G.). - On partitions avoiding 3-crossings. Séminaire Lotharingien de Combinatoire, vol. B54e, pages 1–21.
Gerhold (S.), and Kauers (M.). - A computer proof of Turán's inequality. Journal of Inequalities in Pure and Applied Mathematics, vol. 7, n°2, Article 42.
Kauers (M.). - Shift equivalence of P-finite sequences. The Electronic Journal of Combinatorics, vol. 13, R. 100, pages 1–16.
Kovacs (L.), Popov (N.), and Jebelean (T.). - Combining logic and algebraic techniques for program verification in theorema. Second International Symposium on Leveraging Applications of Formal Methods, Verification and Validation, IEEE Press, pages 67–74.
Kovacs (L.), Popov (N.), and Jebelean (T.). - Experimental Program Verification in the Theorema System. Int. Journal on Software Tools for Technology Transfer.
Massazza (P.) and Radicioni (R.) - A symbolic approach to computing the holonomic functions. Pure Mathematics and Applications vol. 17, n°1-2, 147–164.
Scott (T. C.), Aubert-Frécon (M.), and Grotendorst (J.). - New approach for the electronic energies of the hydrogen molecular ion. Chemical Physics, vol. 324, n°2-3, pages 323–338.
Rechnitzer, (A.). - Haruspicy 3: The anisotropic generating function of directed bond-animals is not D-finite. Journal of Combinatorial Theory. Series A, vol. 113, n°6, pages 1031–1049.

2005

Andrews (G. E.), Paule (P.) and Schneider (C.) - Plane partitions VI: Stembridge's TSPP theorem. Advances in Applied Mathematics, vol. 34, n°4, pages 709–739.
Bousquet-Mélou. - Algebraic generating functions in enumerative combinatorics and context-free languages. STACS 2005, Lecture Notes in Computer Science, vol. 3404, pages 18–35.
Chyzak (F.), Mishna (M.), and Salvy (B.). - Effective scalar products of D-finite symmetric functions. Journal of Combinatorial Theory, Series A, vol. 112, n°1, October 2005, pages 1–43.
Kovacs (L.) and Jebelean (T.). - Automated generation of loop invariants by recurrence solving in theorema. Proceedings of the 6th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC04), Timisoara, Romania, Analele Universitatii din Timisoara, Vol. XXXIX, Fasc. 2, Seria Matematica-Informatica.
Ulmer (F.) - Note on algebraic solutions of differential equations with known finite Galois group. Applicable Algebra in Engineering, Communications and Computing, vol. 16, n°4, pages 205–218.

2004

Ahmed (M.), Ward (R.) and Kharma (N.) - Solving mathematical problems using knowledge-based systems. Mathematics and Computers in Simulation, vol. 67, n°1-2, pages 149–161.
Foupouagnigni (M.), Koepf (W.) and Ronveaux (A.). - Factorization of fourth-order differential equations for perturbed classical orthogonal polynomials. Journal of Computational and Applied Mathematics, vol. 162, n°2, pages 299–326.
Schneider (C.). - The Summation Package Sigma: Underlying Principles and a Rhombus Tiling Application. Discrete Mathematics and Theoretical Computer Science, vol. 6, n°2, pages 365–386.

2003

Barkatou (M.), Chyzak (F.), and Loday-Richaud (M.). - Remarques algorithmiques liées au rang d'un opérateur différentiel linéaire. IRMA Lectures in Mathematics and Theoretical Physics, vol. 3, de Gruyter, pages 87–129.
Brennan (C. A. C.) and Prodinger (H.). - The pills problem revisited. Quaestiones Mathematicae, vol. 26, n°4, pages 427–439.
Durand (M.). - Asymptotic analysis of an optimized quicksort algorithm. Inf. Process. Lett., vol. 85, n°2, 2003, pages 73–77.
Foupouagnigni (M.), Koepf (W.) and Ronveaux (A.). - On fourth-order difference equations for orthogonal polynomials of a discrete variable: Derivation, factorization and solutions. Journal of Difference Equations and Applications, vol. 9, n°9, pages 777–804.
Koepf (W.) and VanAssche (W.) - Computer algebra algorithms for orthogonal polynomials and special functions. Orthogonal polynomials and special functions, Lecture notes in Mathematics, vol. 1817, pages 1–24.
Meunier (L.) and Salvy (B.). - ESF: An automatically generated encyclopedia of special functions. ISSAC'03, ACM Press, pages 199–205.
Paule (P.) and Schneider (C.). - Computer proofs of a new family of harmonic number identities. Advances in Applied Mathematics, vol. 31, n°2, 2003, pages 359–378.
Sloane (N. J. A.) - The on-line encyclopedia of integer sequences. Notices of the American Mathematical Society, vol. 50, n°8, pages 912–915.

2002

Banderier (C.) and Flajolet (P.). - Basic analytic combinatorics of directed lattice paths. Theor. Comput. Sci., vol. 281, n°1-2, 2002, pages 37–80.
Deutsch (E.) and Noy (M.). - Statistics on non-crossing trees. Discret. Math., vol. 254, n°1-3, 2002, pages 75–87.
Fabijonas (B. R.). - Laplace's method on a computer algebra system with an application to the real valued modified bessel functions. J. Comput. Appl. Math., vol. 146, n°2, 2002, pages 323–342.
Flajolet (P.), Hatzis (K.), Nikoletseas (S.), and Spirakis (P.). - On the robustness of interconnections in random graphs: a symbolic approach. Theoretical Computer Science, vol. 287, n°2, 2002, pages 515–534.
Guttmann (A. J.) and Vöge (M.). - Lattice paths: vicious walkers and friendly walkers. Journal of Statistical Planning and Inference vol. 101, n°1-2, pages 107–131.
Nicodème (P.), Salvy (B.), and Flajolet (P.). - Motif statistics. Theoretical Computer Science, vol. 287, n°2, 2002, pages 593–617.
Van der Hoeven (J.). - Relax, but don't be too lazy. Journal of Symbolic Computation, vol. 34, n°6, 2002, pages 479–542.

2000

Beckermann (B.) and Labahn (G.). - Fraction-free computation of matrix rational interpolants and matrix gcds. SIAM J. Matrix Anal. Appl., vol. 22, n°1, 2000, pages 114–144.
Beckermann (B.) and Labahn (G.). - Effective computation of rational approximants and interpolants. Reliable Computing vol. 6, n°4, pages 365–390.
Bergeron (F.) and Gascon (F.). - Counting Young tableaux of bounded height. Journal of Integer Sequences vol. 3, n°1, Article 00.1.7.
Bousquet-Mélou (M.). - Sorted and/or sortable permutations. Discret. Math., vol. 225, n°1-3, 2000, pages 25–50.
Brogelli (S.), Merlini (D.) and Sprugnoli (R.) - Proving a class of hypergeometric identities by means of generating functions Formal Power Series and Algebraic Combinatorics, 12th FPSAC conference, Moscow, Russia, pages 645–654.
Deutsch (E.) and Noy (M.) - New statistics on non-crossing trees. Formal Power Series and Algebraic Combinatorics, 12th FPSAC conference, Moscow, Russia, pages 667–676.
Fulmek (M.) and Krattenthaler (C.), - The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, II. Europ. J. Combinatorics, vol. 21, pages 601–640.
Lercier (R.) and Morain (F.). - Computing isogenies between elliptic curves over $F_p^n$ using {C}ouveignes's algorithm. Math. Comput., vol. 69, n°229, 2000, pages 351–370.
Litow (B.). - On Hadamard square roots of unity. Theoretical Computer Science, vol. 237, n°1-2, 2000, pages 447–454.
Suter (R.). - Two analogues of a classical sequence. Journal of Integer Sequences, vol. 3, n°1.

1999

Chyzak (F.), Gutman (I.), and Paule (P.). - Predicting the number of hexagonal systems with 24 and 25 hexagons. Match-Commun. Math. Chem., n°40, 1999, pages 139–151.
Denise (A.) and Zimmermann (P.). - Uniform random generation of decomposable structures using floating-point arithmetic. Theoretical Computer Science, vol. 218, n°2, pages 233–248.
Evans (R.), Hollmann (H. D. L.), Krattenthaler (C.) and Xiang (Q.). - Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets. Journal of Combinatorial Theory, series A, vol. 87, n°1, pages 74–119.
Flajolet (P.) and Noy (M.). - Analytic combinatorics of non-crossing configurations. Discret. Math., vol. 204, n°1-3, 1999, pages 203–229.
Krattenthaler (C.). - Advanced determinant calculus Séminaire Lotharingien de Combinatoire, vol. 42, pages 1–67.
Ronveaux (A.), Zarzo (A.), Area (I.), and Godoy (E.). - Decomposition of polynomials with respect to the cyclic group of order m. Journal of Symbolic Computation, vol. 28, n°6, 1999, pages 755–765.

1998

Bousquet-Mélou (M.). - Multi-statistic enumeration of two-stack sortable permutations. Electronic Journal of Combinatorics, vol. 5, n°1, pages 1–12.
Chyzak (F.) and Salvy (B.). - Non-commutative elimination in Ore algebras proves multivariate identities. Journal of Symbolic Computation, vol. 26, n°2, 1998, pages 187–227.
Denise (A.), Dutour (I.), and Zimmermann (P.). - CS: a Mupad package for counting and randomly generating combinatorial structures. Formal Power Series and Algebraic Combinatorics, pages 1–10.
Krattenthaler (C.). - An alternative evaluation of the Andrews-Burge determinant. Progress in Mathematics, vol. 161., pages 263–270.
Martínez (C.), Panholzer (A.), and Prodinger (H.). - On the number of descendants and ascendants in random search trees. Electronic Journal of Combinatorics vol. 5, n°1, pages 1–26.
Panholzer (A.) and Prodinger (H.). - Towards a more precise analysis of an algorithm to generate binary trees: A tutorial. Comput. J., vol. 41, n°3, 1998, pages 201–204.
Prodinger (H.). - Combinatorial sums in the analysis of a generalized van der Pol equation. Mapletech, vol. 5, n°1, 1998, pages 33–36.

1997

Abramov (S. A.) and Zima (E. V.). - Minimal completely factorable annihilators. ISSAC'97, ACM Press, pages 290–297.
Bauschke (H. H.) and Corless (R. M.). - Analyzing a projection method with maple. Mapletech, vol. 4, n°1, 1997, pages 2–7.
Beckermann (B.), Cabay (S.) and Labahn (G.). - Fraction-free computation of matrix Padé systems. ISSAC'97, ACM Press, pages 125–132.
Borwein (J. M.), Lisonek (P.), and Macdonald (J. A.). - Arithmetic-geometric means revisited. Mapletech, vol. 4, n°1, 1997, pages 20–27.
Bousquet-Mélou (M.) and Guttmann (T.). - Enumeration of three-dimensional convex polygons. Annals of Combinatorics, vol. 1, pages 27–53.
Flajolet (P.) and Salvy (B.). - The Sigsam challenges: Symbolic asymptotics in practice. SIGSAM Bulletin, vol. 31, n°4, pages 36–47.
Koepf (W.). - Identities for families of orthogonal polynomials and special functions. Integral Transforms and Special Functions vol. 5, n°1, pages 69–102.
Li (Z.) and Nemes (I.). - A modular algorithm for computing greatest common right divisors of Ore polynomials. ISSAC'97, ACM Press, 1997, pages 282–289.
Pelz (R. B.) and Gulak (Y.). - Evidence for a real-time singularity in hydrodynamics from time series analysis. Phys. Rev. Lett., vol. 79, n°25, 1997, pages 4998–5001.
Prodinger (H.). - On a question of Cooper and Kennedy. Fibonacci Q., vol. 35, n°2, 1997, pages 135–136.
Tsarev (S. P.). - Symbolic manipulation of integrodifferential expressions and factorization of linear ordinary differential operators over transcendental extensions of a differential field. ISSAC'97, ACM Press, pages 310–315.

1996

Grosse-Kunstleve (R. W.), Brunner (G. O.), and Sloane (N. J. A.). - Algebraic description of coordination sequences and exact topological densities for zeolites. Acta Crystallogr. Sect. A, vol. 52, n°6, 1996, pages 879–889.
Noonan (J.) and Zeilberger (D.) - The Enumeration of Permutations with a Prescribed Number of ``Forbidden'' Patterns. Advances in Applied Mathematics, vol. 17, pages 381–407.
Zeilberger (D.). - Self-avoiding walks, the language of science, and Fibonacci numbers. Journal of Statistical Planning and Inference, vol. 54, pages 135–138.

1995

Bergeron (F.) and Sattler (U.). - Constructible differentially finite algebraic series in several variables. Theoretical Computer Science, vol. 144, n°1-2, pages 59–65.
Flajolet (P.) and Salvy (B.). - Computer algebra libraries for combinatorial structures. Journal of Symbolic Computation, vol. 20, n°5-6, 1995, pages 653–671.
Nemes (I.) and Petkovsek (M.). - Rcomp: A Mathematica package for computing with recursive sequences. Journal of Symbolic Computation, vol. 20, n°5-6, 1995, pages 745–753.
Paule (P.) and Schorn (M.). - A mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. Journal of Symbolic Computation, vol. 20, n°5-6, 1995, pages 673–698.

1994

Strehl (V.). - Binomial identites–combinatorial and algorithmic aspects. Discrete Mathematics, vol. 136, pages 309–346.

1993

Petkovsek (M.) and Salvy (B.). - Finding all hypergeometric solutions of linear differential equations. ISSAC'93, ACM Press, pages 27–33.

1992

Bergeron (F.) and Plouffe (S.) - Computing the Generating Function of a Series given its First Terms. Experimental Mathematics, vol. 1, n°4, pages 308–312.

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