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

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 $\sigma_N=\sqrt{\text{Var}(S_N)}.$