Greatest Factorial Factorization and Symbolic Summation

Abstract

The greatest factorial factorization (GFF) of a polynomial provides an analogue to square-free factorization but with respect to integer shifts instead to multiplicities. We illustrate the fundamental role of that concept in the context of symbolic summation. Besides a detailed discussion of the basic GFF notions we present a new approach to the inde nite rational summation problem as well as to Gosper's algorithm for summing hypergeometric sequences.

Differential Galois theory of linear difference equations

“…Many of these results appear implicitly (and some explicitly) in the literature [1,6,26,29,34,37,42,43,44] but we assemble them here in a form that meets our needs. We begin by considering the field C(x), C algebraically closed, with the difference operator σ(x) = x + 1 or σ(x) = qx, q not a root of unity.…”

Parameterized Telescoping Proves Algebraic Independence of Sums

“…Summarizing, parameterized telescoping in combination with ΠΣ * -fields gives a criterion to check algorithmically the transcendence of sums of type (1.3); see Theorem 5.1. Combining this criterion with results from summation theory, like [Abr71,Pau95,Abr03,Sch07a], shows that whole classes of sequences are transcendental. E.g., the harmonic numbers {H (i) n | i ≥ 1} with H (i) n := n k=1 1 k i are algebraically independent over Q(n).…”

“…The summation criterion from [Abr71], [Pau95,Prop. 3.3] and its generalization to ΠΣ * -extensions are substantial:…”

“…Telescoping [Gos78] and creative telescoping [Zei91,PWZ96] for hypergeometric terms and its variations [PS95,Pau95,PR97,BP99] are standard tools in symbolic summation. All these techniques are covered by the following formulation of the parameterized telescoping problem: Given sequences f 1 (k), .…”

Structural theorems for symbolic summation

“…This framework and extensions [42,43,44,23,48,24,45,25] generalize, e.g., the (q-)hypergeometric algorithms presented in [1,18,54,34,32,35,33,5,20,3], they cover as special case the summation of (q-)harmonic sums [10,51,29,11] arising, e.g., in particle physics, and they can treat classes of multi-sums that are out of scope of, e.g., the holonomic approach [53,52,15,14]. Karr's algorithm can be considered as the discrete analogue of Risch's algorithm [36,37] for indefinite integration.…”