A two-parameter deformation of the quasi-shuffle and new bases of quasi-symmetric functions

TL;DR

This paper introduces a two-parameter deformation of the quasi-shuffle product, leading to new bases of quasi-symmetric functions with a novel product rule derived from a formal group law.

Contribution

It defines a new two-parameter deformation of the quasi-shuffle and constructs corresponding bases of $QSym$ and $ ext{$ extbf{WQSym}$}$.

Findings

New bases of $QSym$ and $ ext{$ extbf{WQSym}$} are constructed.

The product rule for these bases is given by the deformed quasi-shuffle operation.

The deformation is linked to the exponential generating function of Eulerian polynomials.

Abstract

We define a two-parameter deformation of the quasi-shuffle by means of the formal group law associated with the exponential generating function of the homogeneous Eulerian polynomials, and construct bases of $QSym$ and $\WQSym$ whose product rule is given by this operation.