Minimal Representations and Algebraic Relations for Single Nested Products

Abstract

Recently, it has been shown constructively how a finite set of hypergeometric products, multibasic hypergeometric products or their mixed versions can be modeled properly in the setting of formal difference rings. Here special emphasis is put on robust constructions: whenever further products have to be considered, one can reuse -up to some mild modifications-the already existing difference ring. In this article we relax this robustness criteria and seek for another form of optimality. We will elaborate a general framework to represent a finite set of products in a formal difference ring where the number of transcendental product generators is minimal. As a bonus we are able to describe explicitly all relations among the given input products.This work was supported by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15).3 In this example the evaluation of an element from É(ι)(x) is carried out by replacing x with concrete values n ∈ AE. Later we will generalize this simplest case to formal difference rings equipped with an evaluation function acting on the ring elements.4 More generally, if R is a commutative ring with 1, we define the ideal I generated by a 1 , . . . , ar ∈ R with I = a 1 , . . . , ar R = {f 1 a 1 + · · · + fr ar | f 1 , . . . , fr ∈ R}.

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“…Here we will rely on the following property of ΠΣ-ield extensions; the irst statement has been shown in [12] for ΠΣ-ields. The second statement appears also in [25].…”

Order bounds for $C^2$-finite sequences

“…however, only be interpreted as a sequence in K N from some term on (cf. the discussion in Section 8.2 in [27] or [28]). For instance, the sequence 3 n 2 n −1 cannot be evaluated at the term n = 0.…”

“…From the closed form of C-finite sequences it is clear that these sequences can be seen as special cases of sums of single nested product expressions. The torsion number can be used to find a certain algebraic independent basis of these sequences [28].…”

A refined machinery to calculate large moments from coupled systems of linear differential equations

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