  The Schur (resp. Carath\'eodory) class consists of all the analytic functions
$f$ on the unit disk with $|f|\le 1$ (resp. $\Re f>0$ and $f(0)=1$). The Schur
parameters $\gamma_0,\gamma_1,\dots (|\gamma_j|\le 1)$ are known to parametrize
the coefficients of functions in the Schur class. By employing a recursive
formula for it, we describe the $n$-th coefficient of a Carath\'eodory function
in terms of $n$ independent variables $\gamma_1,\dots, \gamma_n$ with
$|\gamma_j|\le 1.$ The mapping properties of those correspondences are also
studied.
