Some Arithmetic Properties of the q-Euler Numbers and q-Salié Numbers

For m>n\geq 0 and 1\leq d\leq m, it is shown that the q-Euler number E_{2m}(q) is congruent to q^{m-n}E_{2n}(q) mod (1+q^d) if and only if m\equiv n mod d. The q-Salié number S_{2n}(q) is shown to be divisible by (1+q^{2r+1})^{\left\lfloor \frac{n}{2r+1}\right\rfloor} for any r\geq 0. Furthermore, similar congruences for the generalized q-Euler numbers are also obtained, and some conjectures are formulated.

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