Congruences modulo cyclotomic polynomials and algebraic independence for q-series

2017/01/23 by Adamczewski, Boris, Bell, Jason P., Delaygue, Éric +1
#05A30 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1701.06378

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.

Related

• A family of q-congruences modulo the square of a cyclotomic polynomial
• A family of q-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial
• A q-analogue of Wilson's congruence
• Some q-congruences arising from certain identities
• Congruences for sums of MacMahon's q-Catalan polynomials