Lagrange inversion formula, Laguerre polynomials and the free unitary Brownian motion

Nizar Demni

Published 2016-06-12Version 1

These notes are devoted to the computations of some relevant quantities related to the free unitary Brownian motion. Using the Lagrange inversion formula, we derive an explicit expression for its alternating star-cumulants of even lengths. Next, we use again this formula and together with a generating series for Laguerre polynomials in order to compute the reciprocal of the free cumulant generating function of the free Jacobi process associated with a single projection of rank $1/2$. Similar computations lead also to the Taylor expansions of the Schur function of the spectral distribution of the free unitary Brownian motion and of its first iterate.