Congruences modulo cyclotomic polynomials and algebraic independence for \(q\)-series. (English. French summary) Zbl 1405.11019

Summary: 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.


11B65 Binomial coefficients; factorials; \(q\)-identities
05A15 Exact enumeration problems, generating functions
05A30 \(q\)-calculus and related topics
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