Algebraic independence of sequences generated by (cyclotomic) harmonic sums

TL;DR

This paper demonstrates that sequences generated by basis sums in the quasi-shuffle algebra are algebraically independent over rational sequences, using difference ring extensions and embedding into sequence rings.

Contribution

It introduces a novel algebraic framework showing the algebraic independence of sequences from (cyclotomic) harmonic sums via difference ring embeddings.

Findings

Sequences from basis sums are algebraically independent.

Embedding difference rings into sequence rings preserves algebraic independence.

Constants remain unchanged in the difference ring extensions.

Abstract

An expression in terms of (cyclotomic) harmonic sums can be simplified by the quasi-shuffle algebra in terms of the so-called basis sums. By construction, these sums are algebraically independent within the quasi-shuffle algebra. In this article we show that the basis sums can be represented within a tower of difference ring extensions where the constants remain unchanged. This property enables one to embed this difference ring for the (cyclotomic) harmonic sums into the ring of sequences. This construction implies that the sequences produced by the basis sums are algebraically independent over the rational sequences adjoined with the alternating sequence.