ABSTRACT
New algorithms are presented for computing the dispersion set of two polynomials over Q and for shiftless factorization. Together with a summability criterion by Abramov, these are applied to get a polynomial-time algorithm for indefinite rational summation, using a sparse representation of the output.
- S.A. Abramov. On the summation of rational functions. U.S.S.R. Comput. Maths. Math. Phys. 11, pp. 324--330, 1971. Transl. from Zh. vychisl. mat. mat. fiz. 11, pp. 1071--1075, 1971.Google ScholarCross Ref
- S.A. Abramov. The rational component of the solution of a first-order linear recurrence relation with a rational right-hand side. U.S.S.R. Comput. Maths. Math. Phys. 15, pp. 216--221, 1975. Transl. from Zh. vychisl. mat. mat. fiz. 15, pp. 1035--1039, 1975.Google ScholarCross Ref
- S.A. Abramov. Indefinite sums of rational functions. Proceedings ISSAC'95, pp. 303--308. Google ScholarDigital Library
- E. Bach, J. Driscoll and J. O. Shallit. Factor Refinement. Journal of Algorithms. 15, pp. 199--222, 1993. Google ScholarDigital Library
- E. Bach and J. Shallit. Algorithmic Number Theory. Volume 1: Efficient Algorithms. MIT Press, Boston MA, 1996. Google ScholarDigital Library
- J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, Cambridge, U.K., 1999. Google ScholarDigital Library
- J. Gerhard. Modular algorithms in symbolic summation and symbolic integration. PhD thesis, Universität Paderborn, Germany, 2001.Google Scholar
- R.W. Gosper. Decision procedures for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. U.S.A. 75(1), pp. 40--42, 1978.Google ScholarCross Ref
- M. van Hoeij. Rational solutions of linear difference equations. Proceedings ISSAC'98, pp. 120--123, 1998. Google ScholarDigital Library
- M. van Hoeij. Factoring polynomials and the knapsack problem. J. Number Theory. 95, pp. 167-189, 2002.Google ScholarCross Ref
- P. Lisonek, P. Paule and V. Strehl. Improvement of the Degree Setting in Gosper's Algorithm. J. Symbolic Comput. 16, pp. 243--258, 1993. Google ScholarDigital Library
- Y.K Man. On computing closed forms for indefinite summation. J. Symbolic Comput. 16, pp. 355--376, 1993. Google ScholarDigital Library
- Y.K. Man and F.J. Wright. Fast polynomial dispersion computation and its application to indefinite summation. Proceedings ISSAC'94, pp. 175--180, 1994. Google ScholarDigital Library
- J.B. Rosser and L. Schoenfeld. Approximate formulas for some functions of prime numbers. Ill. J. Math. 6, pp. 64--94, 1962.Google ScholarCross Ref
- P. Paule. Greatest factorial factorization and symbolic summation. J. Symbolic Comput. 20(3), pp. 235--268, 1995. Google ScholarDigital Library
- R. Pirastu. On combinatorial identities: symbolic summation and umbral calculus. PhD thesis, Johannes Kepler Universität Linz, Austria, July 1996.Google Scholar
Index Terms
- Shiftless decomposition and polynomial-time rational summation
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