  We obtain asymptotic expansions for probabilities $\mathbb{P}(S_N=k)$ of
partial sums of uniformly bounded integer-valued functionals $S_N=\sum_{n=1}^N
f_n(X_n)$ of uniformly elliptic inhomogeneous Markov chains. The expansions
involve products of polynomials and trigonometric polynomials, and they hold
without additional assumptions. As an application of the explicit formulas of
the trigonometric polynomials, we show that for every $r\geq1\,$, $S_N$ obeys
the standard Edgeworth expansions of order $r$ in a conditionally stable way if
and only if for every $m$, and every $\ell$ the conditional distribution of
$S_N$ given $X_{j_1},...,X_{j_\ell}$ mod $m$ is $o_\ell(\sigma_N^{1-r})$ close
to uniform, uniformly in the choice of $j_1,...,j_\ell$, where
$\sigma_N=\sqrt{\text{Var}(S_N)}.$
