## 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.

### MSC:

 11B65 Binomial coefficients; factorials; $$q$$-identities 05A15 Exact enumeration problems, generating functions 05A30 $$q$$-calculus and related topics
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### References:

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