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: ∏k = n0n − 1R(k) = F(n)∏k = n0n − 1V(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 ∏k = n0n − 1R(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 “non-summable part" of T(n). In other words, we get a minimal additive decomposition of the hypergeometric term T(n).