Analytic aspects of the shuffle product

TL;DR

This paper explores the combinatorial and algebraic properties of the shuffle product, demonstrating its relevance to D-finite generating functions and introducing a new grammar class that models these functions.

Contribution

It introduces a new grammar class that models D-finite generating functions and analyzes the shuffle product's role in understanding algebraic and rational functions.

Findings

Shuffle product models key aspects of D-finite generating functions

Explicit generating function consequences are derived

A new grammar class for D-finite functions is defined

Abstract

There exist very lucid explanations of the combinatorial origins of rational and algebraic functions, in particular with respect to regular and context free languages. In the search to understand how to extend these natural correspondences, we find that the shuffle product models many key aspects of D-finite generating functions, a class which contains algebraic. We consider several different takes on the shuffle product, shuffle closure, and shuffle grammars, and give explicit generating function consequences. In the process, we define a grammar class that models D-finite generating functions.