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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
##### Keywords:
$$q$$-analogues; congruences
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##### References:
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