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

Source abstract

We study two linear bases of the free associative algebra ℤ 〈 X , Y 〉 : one is formed by the Magnus polynomials of type ( ad X k 1 Y ) ⋯ ( ad X k d 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 ℤ 〈 X , Y 〉 . As an application, we derive a formula of Le–Murakami, Furusho type that expresses arbitrary coefficients of a group-like series J ∈ ℂ 〈 〈 X , Y 〉 〉 in terms of the “regular” coefficients of J .