For m > n ≥ 0 and 1 ≤ d ≤ m, it is shown that the q-Euler number E2m(q) is congruent to qm-n E2n(q) mod (1 + qd) if and only if m ≡ n mod d. The q-Salie number S2n(q) is shown to be divisible by (1 + q2r+1)⌊n/2r+1⌋ for any r ≥ 0. Furthermore, similar congruences for the generalized q-Euler numbers are also obtained, and some conjectures are formulated.
['Victor J. W. Guo', 'Jiang Zeng']
Some arithmetic properties of the q -Euler numbers and q -Salié numbers