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

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For m>n≥ 0 and 1≤ d≤ m, it is shown that the q-Euler number E₂ₘ(q) is congruent to qᵐ⁻ⁿE₂ₙ(q) mod (1+qᵈ) if and only if m≡ n mod d. The q-Salié 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.