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Séminaire Lotharingien de Combinatoire, 78B.54 (2017), 12 pp.

# Boris Adamczewski, Jason P. Bell,
Éric Delaygue and Frédéric Jouhet

# Congruences Modulo Cyclotomic Polynomials and Algebraic Independence for *q*-Series

**Abstract.**
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.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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