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

For m > n ≥ 0 and 1 ≤ d ≤ m , it is shown that the q -Euler number E 2 m ( q ) is congruent to q m − n E 2 n ( q ) mod ( 1 + q d ) if and only if m ≡ n mod d . The q -Salié number S 2 n ( q ) is shown to be divisible by ( 1 + q 2 r + 1 ) ⌊ n 2 r + 1 ⌋ for any r ≥ 0 . Furthermore, similar congruences for the generalized q -Euler numbers are also obtained, and some conjectures are formulated.