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

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

2005-04-28

arxiv.org/abs/math/0504569v2

European J. Combin. 27 (2006), 884--895

Mathematics

Combinatorics

12 pages, see also http://math.univ-lyon1.fr/~guo

Scientific paper

10.1016/j.ejc.2005.04.009

For m>n\geq 0 and 1\leq d\leq m, it is shown that the q-Euler number E_{2m}(q) is congruent to q^{m-n}E_{2n}(q) mod (1+q^d) if and only if m\equiv n mod d. The q-Sali\'e number S_{2n}(q) is shown to be divisible by (1+q^{2r+1})^{\left\lfloor \frac{n}{2r+1}\right\rfloor} for any r\geq 0. Furthermore, similar congruences for the generalized q-Euler numbers are also obtained, and some conjectures are formulated.

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