Some arithmetic properties of the qq-Euler numbers and qq-Salié numbers

For m>n≥0m>n≥0 and 1≤d≤m1≤d≤m, it is shown that the qq-Euler number E2m(q)E2m(q) is congruent to qm−nE2n(q)mod(1+qd)qm−nE2n(q)mod(1+qd) if and only if m≡nmoddm≡nmodd. The qq-Salié number S2n(q)S2n(q) is shown to be divisible by (1+q2r+1)⌊n2r+1⌋ for any r≥0r≥0. Furthermore, similar congruences for the generalized qq-Euler numbers are also obtained, and some conjectures are formulated.