Rational normal forms and minimal decompositions of hypergeometric terms

  • Authors:

  • S. A. Abramov;M. Petkovsek

  • Affiliations:

  • Dorodnicyn Computing Centre, Russian Academy of Science, Moscow, Russia;Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia

  • Venue:
  • Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
  • Year:
  • 2002

Quantified Score

Hi-index 0.00

Visualization

Abstract

We describe a multiplicative normal form for rational functions which exhibits the shift structure of the factors, and investigate its properties. On the basis of this form we propose an algorithm which, given a rational function R, extracts a rational part F from the product of consecutive values of R: Πk=n0n-1 R(k) =F(n)Πk=n0n-1 V(k) where the numerator and denominator of the rational function V have minimal possible degrees. This gives a minimal multiplicative representation of the hypergeometric term Πk=n0n-1 R(k). We also present an algorithm which, given a hypergeometric term T(n), constructs hypergeometric terms T1(n) and T2(n) such that T(n) = ΔT1(n) + T2(n) and T2(n) is minimal in some sense. This solves the additive decomposition problem for indefinite sums of hypergeometric terms: ΔT1(n) is the "summable part", and T2(n) the "nonsummable part" of T(n). In other words, we get a minimal additive decomposition of the hypergeometric term T(n).