A one-parameter family of dendriform identities

TL;DR

This paper introduces a new q-identity within the dendriform dialgebra of colored free quasi-symmetric functions, generalizing known identities and including the noncommutative Bohnenblust-Spitzer identity as a special case.

Contribution

It establishes a one-parameter family of identities in dendriform algebras, extending previous results and unifying several important identities.

Findings

Proves a q-identity in dendriform dialgebra

Recovers known identities at q=1

Generalizes the noncommutative Bohnenblust-Spitzer identity

Abstract

We prove a q-identity in the dendriform dialgebra of colored free quasi-symmetric functions. For q=1, we recover identities due to Ebrahimi-Fard, Manchon, and Patras, in particular the noncommutative Bohnenblust-Spitzer identity.