Algorithmic Manipulations and Transformations of Univariate Holonomic Functions and Sequences

Holonomic functions and sequences have the property that they can be represented by a finite amount of information. Moreover, these holonomic objects are closed under elementary operations like, for instance, addition or (termwise and Cauchy) multiplication. These (and other) operations can also be performed “algorithmically”. As a consequence, we can prove any identity of holonomic functions or sequences automatically. Based on this theory, the author implemented a package that contains procedures for automatic manipulations and transformations of univariate holonomic functions and sequences within the computer algebra system Mathematica . This package is introduced in detail. In addition, we describe some different techniques for proving holonomic identities.

[1]  Marko Petkovsek,et al.  Hypergeometric Solutions of Linear Recurrences with Polynomial Coefficents , 1992, J. Symb. Comput..

[2]  D. Zeilberger.  A holonomic systems approach to special functions identities , 1990 .

[4]  Marko Petkovsek,et al.  RComp: A Mathematica Package for Computing with Recursive Sequences , 1995, J. Symb. Comput..

[5]  Ziming Li,et al.  A modular algorithm for computing greatest common right divisors of Ore polynomials , 1997, ISSAC.

[6]  Manuel Bronstein,et al.  On ore rings, linear operators and factorisation , 1993 .

[7]  Richard P. Stanley,et al.  Differentiably Finite Power Series , 1980, Eur. J. Comb..

[8]  Ira M. Gessel,et al.  Symmetric functions and P-recursiveness , 1990, J. Comb. Theory, Ser. A.

[9]  E. Hansen.  A Table of Series and Products , 1977 .

[10]  Bruno Salvy,et al.  Non-Commutative Elimination in Ore Algebras Proves Multivariate Identities , 1998, J. Symb. Comput..

[11]  Frédéric Chyzak,et al.  Holonomic systems and automatic proofs of identities , 1994 .

[12]  L. Lipshitz,et al.  D-finite power series , 1989 .

[13]  O. Ore.  Theory of Non-Commutative Polynomials , 1933 .

[14]  Daniel Richardson,et al.  Some undecidable problems involving elementary functions of a real variable , 1969, Journal of Symbolic Logic.

[15]  Wolfram Koepf,et al.  Power Series in Computer Algebra , 1992, J. Symb. Comput..

[16]  Joseph F. Traub,et al.  On Euclid's Algorithm and the Theory of Subresultants , 1971, JACM.

[17]  Reinwald Jungen.  Sur les séries de Taylor n'ayant que des singularités algébrico-logarithmiques sur leur cercle de convergence , 1931 .

[18]  Peter Paule,et al.  A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coefficient Identities , 1995, J. Symb. Comput..

[19]  François Bergeron,et al.  Computing the Generating Function of a Series Given Its First Few Terms , 1992, Exp. Math..

[20]  Philippe Flajolet,et al.  The Average case analysis of algorithms : counting and generating functions , 1993 .

[21]  Doron Zeilberger,et al.  Sister Celine's technique and its generalizations , 1982 .