# Journal of Operator Theory

Volume 78, Issue 1,  Summer  2017  pp. 179-200.

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

Authors:  Nizar Demni
Author institution: Universite de Rennes, Rennes, 35042, France

Summary:  This paper is devoted to the computations of some relevant quantities associated with the free unitary Brownian motion. Using the Lagrange inversion formula, we first derive an explicit expression for its alternating star cumulants which have even lengths and relate them to those having odd lengths by means of a summation formula for the free cumulants with product as entries. Next, we use again Lagrange formula together with a generating series for Laguerre polynomials in order to compute the Taylor coefficients of the reciprocal of the $R$-transform of the free Jacobi process associated with a single projection of rank $1/2$, and those of its $S$-transform as well. This generating series 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.

DOI: http://dx.doi.org/10.7900/jot.2016jun19.2139
Keywords:  free unitary Brownian motion, alternating star cumulants, free Jacobi process, Lagrange inversion formula, Laguerre polynomials, Verblunsky coefficients