Some Arithmetic Properties of the q-Euler Numbers and q-Sali\'e Numbers
Abstract
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-Sali\'e number S2n(q) is shown to be divisible by (1+q2r+1) n2r+1 for any r≥ 0. Furthermore, similar congruences for the generalized q-Euler numbers are also obtained, and some conjectures are formulated.
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