Greatest Factorial Factorization and Symbolic Summation

Abstract The greatest factorial factorization (GFF) of a polynomial provides an analogue to square-free factorization but with respect to integer shifts instead to multiplicities. We illustrate the fundamental role of that concept in the context of symbolic summation. Besides a detailed discussion of the basic GFF notions we present a new approach to the indefinite rational summation problem as well as to Gosper's algorithm for summing hypergeometric sequences.

[1]  Ronald L. Graham,et al.  Concrete mathematics - a foundation for computer science , 1991 .

[2]  R. W. Gosper.  Decision procedure for indefinite hypergeometric summation. , 1978, Proceedings of the National Academy of Sciences of the United States of America.

[3]  D. E. G. Malm,et al.  The Summation of Rational Functions by an Extended Gosper Algorithm , 1995, J. Symb. Comput..

[4]  S. Abramov.  On the summation of rational functions , 1971 .

[5]  Peter Paule,et al.  SYMBOLIC SUMMATION SOME RECENT DEVELOPMENTS , 1995 .

[6]  R. Moenck.  On computing closed forms for summations. [polynomials and rational functions] , 1977 .

[7]  V. Strehl,et al.  Improvement of the Degree Setting in Gosper's Algorithm Improvement of the Degree Setting in Gosper's Algorithm , 1993 .

[8]  Roberto Pirastu,et al.  Parallel Computation and Indefinite Summation: A MAPLE Application for the Rational Case , 1995, J. Symb. Comput..

[9]  Petr Lisonek,et al.  Improvement of the Degree Setting in Gospers's Algorithm , 1993, J. Symb. Comput..

[10]  James H. Davenport,et al.  Computer Algebra: Systems and Algorithms for Algebraic Computation , 1988 .

[11]  Michael Karr,et al.  Theory of Summation in Finite Terms , 1985, J. Symb. Comput..

[12]  S. A. Abramov,et al.  The rational component of the solution of a first-order linear recurrence relation with a rational right side , 1975 .

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

[14]  E. Horowitz.  Algorithms for partial fraction decomposition and rational function integration , 1971, SYMSAC '71.

[15]  Barry M. Trager,et al.  Algebraic factoring and rational function integration , 1976, SYMSAC '76.

[16]  T. N. Subramaniam,et al.  How to integrate rational functions , 1992 .

[17]  Keith O. Geddes,et al.  Algorithms for computer algebra , 1992 .

[18]  Roberto PIRASTU,et al.  Parallel Computation and Indeenite Summation: a Kmaplek Application for the Rational Case Parallel Computation and Indeenite Summation: a Kmaplek Application for the Rational Case , 1995 .

[19]  Michael Rothstein.  Aspects of symbolic integration and simplification of exponential and primitive functions. , 1976 .

[20]  Markus Schorn,et al.  Contributions to Symbolic Summation , 1995 .

[21]  Sergei A. Abramov.  Rational solutions of linear difference and q-difference equations with polynomial coefficients , 1995, ISSAC '95.

[22]  Michael Karr,et al.  Summation in Finite Terms , 1981, JACM.

[23]  Roberto Pirastu,et al.  Algorithms for Indeenite Summation of Rational Functions in Maple Algorithms for Indeenite Summation of Rational Functions in Maple , 2007 .

[24]  Roberto Pirastu,et al.  Rational Summation and Gosper-Petkovsek Representation , 1995, J. Symb. Comput..