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Some arithmetic properties of the \(q\)-Euler numbers and \(q\)-Salié numbers. (English) Zbl 1111.05009
The authors study a \(q\)-analogue \(E_{2n}(q)\) of the Euler numbers defined by \[ \sum_{n=0}^{\infty}E_{2n}(q)\frac{x^{2n}}{(q;q)_{2n}}=\left( \sum_{n=0}^{\infty}\frac{x^{2n}}{(q;q)_{2n}} \right)^{-1}. \] They show that \(E_{2m}(q)\) is congruent to \(q^{m-n}E_{2n}(q) \mod (1+q^d)\) if and only if \(m\equiv n \pmod d\). Divisibility properties of \(q\)-Salié numbers are also studied.

MSC:
11B65 Binomial coefficients; factorials; \(q\)-identities
05A30 \(q\)-calculus and related topics
05A15 Exact enumeration problems, generating functions
11A07 Congruences; primitive roots; residue systems
11B68 Bernoulli and Euler numbers and polynomials
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