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We consider a family of polynomial analogs of the Ap\'ery numbers, whichincludes $q$-analogs of Krattenthaler--Rivoal--Zudilin and Zheng, and show thatthe supercongruences that Gessel and Mimura established for the Ap\'ery numbersgeneralize to these polynomials. Our proof relies on polynomial analogs ofclassical binomial congruences of Wolstenholme and Ljunggren. We furtherindicate that this approach generalizes to other supercongruence results.

Supercongruences for polynomial analogs of the Ap\'ery numbers