Rational Normal Forms and Minimal Decompositions of Hypergeometric Terms

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: ∏n−1 k=n0 R(k) = F (n) ∏n−1 k=n0 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 ∏n−1 k=n0 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) = ∆T1(n) + T2(n) and T2(n) is minimal in some sense. This solves the additive decomposition problem for indefinite sums of hypergeometric terms: ∆T1(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). c © 2002 Elsevier Science Ltd. All rights reserved.