Asymptotic expansions in the central limit theorem

Let $x_1,x_2,\dots$ be a sequence of independent identically distributed random variables with zero means and unit variances. Put
$$ F_n(x)=\mathbf P\{(x_1+\dots+x_n)/\sqrt n<x\}. $$

Conditions are given which are necessary and sufficient for the relation
$$ F_n(x)=\sum_{\nu=0}^{s-2}n^{-\nu/2}f_\nu(x)+O(\varepsilon_n),\quad n\to\infty, $$
to hold uniformly in $x$, where $s\ge2$, the sequence $\varepsilon_n$ is such that
$$ \varepsilon_nn^{(s-2)/2}\to0,\quad\varepsilon_n\ge n^{-(s-1)/2},\quad n\to\infty, $$
the functions $t_\nu(x)$ are independent of $n$ and satisfy some conditions at the origin.
We consider also local limit theorems.