Greatest Factorial Factorization and Symbolic Summation

At present the most general algebraic and algorithmic frame for discussing the problem of indefinite summation is provided by the work of Karr (1981, 1985). His method, working over ΠΣ-fields which are certain difference field extensions of a constant field K, can be viewed as a summation analogue to Risch’s integration method. A difference field simply is a field F together with an automorphism σ of F. Given a, f from a ΠΣ-field F, Karr’s method constructively decides the existence of a solution g ∈ E of σg − a · g = f where E is a fixed ΠΣ-extension field of F; f is called summable (with respect to E) if the equation can be solved in the case a = 1. We distinguish two cases: The “telescoping problem”, i.e., given f ∈ F find g ∈ F such that σg− g = f , and the “general problem”, i.e., given f ∈ F determine a ΠΣ-extension field E of F such that σg − g = f for some g ∈ E. Despite the fact that Karr’s method algorithmically decides whether a proposed extension E is a ΠΣ-extension, the problem with applying Karr’s method for the general case consists in finding an appropriate candidate for E. But in view of his analogue to Liouville’s theorem on elementary integrals (Karr, 1985, RESULT p. 314) one has the following: If f ∈ F is summable in E, then the “interesting” part of it already is summable in F, and the remainder consists of formal sums that have been adjoined to F in the construction of E. This justifies to consider the telescoping problem separately. Pointing to rational and hypergeometric summation techniques which do not require complete factorization, Karr (1981, sect. 4.2) raises the question whether similar techniques can be “profitably applied” in his ΠΣ-field theory. Despite the fact that the present