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

Boris Adamczewski (1, 2) , Jason P. Bell (3) , Eric Delaygue (1, 2) , Frederic Jouhet (1, 2)

1 ICJ - Institut Camille Jordan
2 CTN - Combinatoire, théorie des nombres
3 SFU - Department of Mathematics [Burnaby]

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.

Domaines: Combinatoire [math.CO]; Théorie des nombres [math.NT]

Boris Adamczewski, Jason P. Bell, Eric Delaygue, Frederic Jouhet. Congruences modulo cyclotomic polynomials and algebraic independence for $q$-series. Seminaire Lotharingien de Combinatoire, 2017, ⟨10.48550/arXiv.1701.06378⟩. ⟨hal-02065319⟩