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This paper is devoted to the computations of some relevant quantitiesassociated with the free unitary Brownian motion. Using the Lagrange inversionformula, we first derive an explicit expression for its alternating starcumulants of even lengths and relate them to those having odd lengths by meansof a summation formula for the free cumulants with product as entries. Next, weuse again this formula together with a generating series for Laguerrepolynomials in order to compute the Taylor coefficients of the reciprocal ofthe $R$-transform of the free Jacobi process associated with a singleprojection of rank $1/2$ and those of the $S$-transform as well. Thisgenerating series lead also to the Taylor expansions of the Schur function ofthe spectral distribution of the free unitary Brownian motion and of its firstiterate.

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