The mechanics of shuffle products and their siblings

TL;DR

This paper explores a new class of shuffle products called $$-shuffle products, analyzing their combinatorial properties, algebraic structures, and conditions for bi-algebra construction, extending Radford's theorem.

Contribution

It introduces and studies the class of $$-shuffle products, extending Radford's theorem and providing conditions for their bi-algebra structures from a combinatorial perspective.

Findings

Defined $$-shuffle products and grouped them with known products

Extended Radford's theorem to this new class of products

Identified conditions for constructing bi-algebras from these products

Abstract

We carry on the investigation initiated in [15] : we describe new shuffle products coming from some special functions and group them, along with other products encountered in the literature, in a class of products, which we name $\varphi$-shuffle products. Our paper is dedicated to a study of the latter class, from a combinatorial standpoint. We consider first how to extend Radford's theorem to the products in that class, then how to construct their bi-algebras. As some conditions are necessary do carry that out, we study them closely and simplify them so that they can be seen directly from the definition of the product. We eventually test these conditions on the products mentioned above.