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Extended rate, more GFUN. (English) Zbl 1218.68204
Summary: We present a software package that guesses formulae for sequences of, for example, rational numbers or rational functions, given the first few terms. We implement an algorithm due to Bernhard Beckermann and George Labahn, together with some enhancements to render our package efficient. Thus we extend and complement Christian Krattenthaler’s program Rate.m, the parts concerned with guessing of Bruno Salvy and Paul Zimmermann’s GFUN, the univariate case of Manuel Kauers’ Guess.m and Manuel Kauers’ and Christoph Koutschan’s qGeneratingFunctions.m.

68W30 Symbolic computation and algebraic computation
Full Text: DOI arXiv
[1] Abrahams, Paul W., Application of LISP to sequence prediction, ()
[2] Beckermann, Bernhard; Labahn, George, A uniform approach for the fast computation of matrix-type Padé approximants, SIAM journal on matrix analysis and applications, 15, 3, 804-823, (1994), MR1282696 (95f:65030) · Zbl 0805.65008
[3] Beckermann, Bernhard; Labahn, George, Recursiveness in matrix rational interpolation problems, ROLLS symposium (Leipzig, 1996), Journal of computational and applied mathematics, 77, 1-2, 5-34, (1997), MR1440002 (97k:65025) · Zbl 0952.65014
[4] Beckermann, Bernhard; Labahn, George, Fraction-free computation of matrix rational interpolants and matrix gcds, SIAM journal on matrix analysis and applications, 22, 1, 114-144, (2000), (electronic). MR1779720 (2001e:65024) · Zbl 0973.65007
[5] Bergeron, François; Plouffe, Simon, Computing the generating function of a series given its first few terms, Experimental mathematics, 1, 4, 307-312, (1992), MR1257287 (95b:05008) · Zbl 0782.05004
[6] Bostan, Alin; Kauers, Manuel, Automatic classification of restricted lattice walks, Proceedings of the 21st international conference on formal power series and algebraic combinatorics, Discrete mathematics and theoretical computer science, DMTCS, 201-215, (2009) · Zbl 1391.05026
[7] Bostan, Alin, Kauers, Manuel, 2010. (with an Appendix by Mark van Hoeij), The complete generating function for Gessel walks is algebraic, in: Proceedings of the American Mathematical Society. · Zbl 1206.05013
[8] Brak, Richard; Guttmann, Anthony J., Algebraic approximants: a new method of series analysis, Journal of physics. A. mathematical and general, 23, 24, L1331-L1337, (1990), MR1090002 (92c:82065) · Zbl 0723.41017
[9] Dumas, Philippe, 1993. Récurrences mahlériennes, suites automatiques, études asymptotiques, Institut National de Recherche en Informatique et en Automatique (INRIA), Rocquencourt, 1993, Thèse, Université de Bordeaux I, Talence, MR1346304 (96g:11021).
[10] Eğecioğlu, Ömer; Redmond, Timothy; Ryavec, Charles, A multilinear operator for almost product evaluation of Hankel determinants, Journal of combinatorial theory, series A, 117, 1, 77-103, (2010), MR2557881 · Zbl 1227.05031
[11] Fisher, Michael E.; Au-Yang, Helen, Inhomogeneous differential approximants for power series, Journal of physics. A. mathematical and general, 12, 10, 1677-1692, (1979), MR545393 (81d:41005a) · Zbl 0427.41010
[12] Kauers, Manuel, 2005. Algorithms for nonlinear higher order difference equations, Ph.D. Thesis, RISC-Linz, Johannes Kepler University, Linz. http://www.risc.uni-linz.ac.at/people/mkauers/publications/kauers05c.pdf.
[13] Kauers, Manuel, 2009. Guessing Handbook, Tech. Report 2009-07, Johannes Kepler Universität Linz.
[14] Kauers, Manuel, Koutschan, Christoph, 2007. A Mathematica package for \(q\)-holonomic sequences and power series, Tech. Report 2007-16, SFB F013. · Zbl 1180.33030
[15] Kauers, Manuel; Koutschan, Christoph, A Mathematica package for \(q\)-holonomic sequences and power series, The Ramanujan journal, 19, 2, 137-150, (2009), MR2511667 · Zbl 1180.33030
[16] Kauers, Manuel; Koutschan, Christoph; Zeilberger, Doron, Proof of ira gessel’s lattice path conjecture, Proceedings of the national Academy of sciences of the united states of America, 106, 28, 11502-11505, (2009), MR2538821 · Zbl 1203.05010
[17] Klazar, Martin, Bell numbers, their relatives, and algebraic differential equations, Journal of combinatorial theory, series A, 102, 1, 63-87, (2003), MR1970977 (2004d:11014) · Zbl 1017.05021
[18] Krattenthaler, Christian, Advanced determinant calculus, Sém. lothar. combin., 42, 67, (1999), (electronic) · Zbl 0923.05007
[19] Mahler, Kurt, On a class of non-linear functional equations connected with modular functions, Journal of the Australian mathematical society series A, 22, 1, 65-118, (1976), MR0441867 (56 #258) · Zbl 0345.39002
[20] Ostrowski, Alexander, Über dirichletsche reihen und algebraische differentialgleichungen, Mathematische zeitschrift, 8, 3-4, 241-298, (1920), MR1544442 · JFM 47.0292.01
[21] Pivar, Malcolm; Finkelstein, Mark, Automation, using LISP, of inductive inference on sequences, (), 125-136
[22] Prellberg, Thomas; Brak, Richard, Critical exponents from nonlinear functional equations for partially directed cluster models, Journal of statistical physics, 78, 3, 701-730, (1995) · Zbl 1102.82316
[23] Salvy, Bruno; Zimmerman, Paul, GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable, Transactions on mathematical software, 20, 2, 163-177, (1994) · Zbl 0888.65010
[24] Sloane, Neil J.A., A handbook of integer sequences, (1973), Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers] New York, London, MR0357292 (50 #9760)
[25] Sloane, Neil J.A., The on-line encyclopedia of integer sequences, Notices of the American mathematical society, 50, 8, 912-915, (2003), http://www.research.att.com/ njas/sequences. MR1992789 (2004f:11151) · Zbl 1044.11108
[26] Stanley, Richard P., Differentiably finite power series, European journal of combinatorics, 1, 2, 175-188, (1980), MR587530 (81m:05012) · Zbl 0445.05012
[27] Stanley, Richard P., (), With a foreword by Gian-Carlo Rota and Appendix 1 by Sergey Fomin. MR1676282 (2000k:05026)
[28] Tutte, William T., Chromatic sums revisited, Aequationes mathematicae, 50, 1-2, 95-134, (1995), MR1336864 (96i:05070) · Zbl 0842.05031
[29] van der Hoeven, Joris, A new zero-test for formal power series, (), 117-122, (electronic). MR2035239 · Zbl 1072.68707
[30] Zagier, Don, Elliptic modular forms and their applications, (), 1-103, MR2409678 (2010b:11047) · Zbl 1259.11042
[31] Zeilberger, Doron, Dave robbins’ art of guessing, Advances in applied mathematics, 34, 4, 939-954, (2005), MR2129005 · Zbl 1066.01516
[32] Zeilberger, Doron, The holonomic ansatz. I. foundations and applications to lattice path counting, Annals of combinatorics, 11, 2, 227-239, (2007), MR2336017 (2008g:05010) · Zbl 1125.05008
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