Congruences modulo cyclotomic polynomials and algebraic independence for $q$-series Article Swipe

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Congruence relation Mathematics Modulo Factorial Series (stratigraphy) Algebraic number Cyclotomic polynomial Independence (probability theory) Congruence (geometry) Pure mathematics Algebra over a field Combinatorics Discrete mathematics Polynomial Mathematical analysis Paleontology Statistics Geometry Biology

Boris Adamczewski , Jason P. Bell , Éric Delaygue , Frédéric Jouhet ·

YOU? · · 2017 · Open Access · · DOI: https://doi.org/10.48550/arxiv.1701.06378 · OA: W2582882867

We prove congruence relations modulo cyclotomic polynomials for multisums of $q$-factorial ratios, therefore generalizing many well-known $p$-Lucas congruences. Such congruences connect various classical generating series to their $q$-analogs. Using this, we prove a propagation phenomenon: when these generating series are algebraically independent, this is also the case for their $q$-analogs.