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We obtain asymptotic expansions for probabilities $\mathbb{P}(S_N=k)$ ofpartial sums of uniformly bounded integer-valued functionals $S_N=\sum_{n=1}^Nf_n(X_n)$ of uniformly elliptic inhomogeneous Markov chains. The expansionsinvolve products of polynomials and trigonometric polynomials, and they holdwithout additional assumptions. As an application of the explicit formulas ofthe trigonometric polynomials, we show that for every $r\geq1\,$, $S_N$ obeysthe standard Edgeworth expansions of order $r$ in a conditionally stable way ifand only if for every $m$, and every $\ell$ the conditional distribution of$S_N$ given $X_{j_1},...,X_{j_\ell}$ mod $m$ is $o_\ell(\sigma_N^{1-r})$ closeto uniform, uniformly in the choice of $j_1,...,j_\ell$, where$\sigma_N=\sqrt{\text{Var}(S_N)}.$

Edgeworth expansions for integer valued additive functionals of uniformly elliptic Markov chains