Demi-shuffle duals of Magnus polynomials in a free associative algebra

We study two linear bases of the free associative algebra 96(X, Y ): one is formed by the Magnus polynomials of type (adk1X Y ) <middle dot> <middle dot> <middle dot> (adkdX Y ) X k and the other is its dual basis (formed by what we call the "demi-shuffle" polynomials) with respect to the standard pairing on the monomials of 96(X, Y ). As an application, we derive a formula of Le-Murakami, Furusho type that expresses arbitrary coefficients of a group-like series J is an element of C (( X, Y )) in terms of the "regular" coefficients of J .