Rational Normal Forms and Minimal Decompositions of Hypergeometric Terms

✍ Scribed by S.A. Abramov; M. Petkovšek

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 336 KB
• Volume: 33
• Category: Article
• ISSN: 0747-7171
• DOI: 10.1006/jsco.2002.0522

✦ Synopsis

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:

where the numerator and denominator of the rational function V have minimal possible degrees. This gives a minimal multiplicative representation of the hypergeometric term n-1 k=n 0

R(k).

We also present an algorithm which, given a hypergeometric term T (n), constructs hypergeometric terms T 1 (n) and

is minimal in some sense. This solves the additive decomposition problem for indefinite sums of hypergeometric terms: ∆ T 1 (n) is the "summable part", and T 2 (n) the "nonsummable part" of T (n). In other words, we get a minimal additive decomposition of the hypergeometric term T (n).