# 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

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