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We develop generalized approach to obtaining Edgeworth expansions for$t$-statistics of an arbitrary order using computer algebra and combinatorialalgorithms. To incorporate various versions of mean-based statistics, weintroduce Adjusted Edgeworth expansions that allow polynomials in the terms todepend on a sample size in a specific way and prove their validity. Providedresults up to 5th order include one and two-sample ordinary $t$-statistics withbiased and unbiased variance estimators, Welch $t$-test, and moderated$t$-statistics based on empirical Bayes method, as well as general results forany statistic with available moments of the sampling distribution. Theseresults are included in a software package that aims to reach a broad communityof researchers and serve to improve inference in a wide variety of analyticalprocedures; practical considerations of using such expansions are discussed.

General order adjusted Edgeworth expansions for generalized $t$-tests