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

TL;DR

This paper computes key quantities related to free unitary Brownian motion using the Lagrange inversion formula, connecting cumulants, transforms, and spectral distributions through Laguerre polynomials.

Contribution

It introduces explicit formulas for cumulants and transforms of free unitary Brownian motion, linking them with Laguerre polynomials and generating series.

Findings

Explicit expression for alternating star cumulants of even lengths

Relations between even and odd length cumulants via summation formula

Taylor expansions of spectral distribution functions and transforms

Abstract

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 of 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 this 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 the $S$-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.