  We study asymptotic expansions in free probability. In a class of classical
limit theorems Edgeworth expansion can be obtained via a general approach using
sequences of "influence" functions of individual random elements described by
vectors of real parameters $(\varepsilon_1,..., \varepsilon_n)$, that is by a
sequence of functions $h_n(\varepsilon_1,..., \varepsilon_n;t)$,
$|\varepsilon_j| \leq \frac 1 {\sqrt n}$, $j=1,...,n$, $t\in {\mathbb R}$ (or
${\mathbb C}$) which are smooth, symmetric, compatible and have vanishing first
derivatives at zero. In this work we expand this approach to free probability.
As a sequence of functions $h_n(\varepsilon_1,..., \varepsilon_n;t)$ we
consider a sequence of the Cauchy transforms of the sum $\sum_{j=1}^n
\varepsilon_j X_j $, where $(X_j)_{j=1}^n$ are free identically distributed
random variables with nine moments. We derive Edgeworth type expansions for
distributions and densities (under the additional assumption that ${supp} (X_1)
\subset [-\sqrt [3]n, \sqrt [3]n]$) of the sum $\frac 1 {\sqrt n} \sum_{j=1}^n
X_j$ within the interval $(-2,2)$.
