arXiv math/05045692005-04math.CO

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

Victor J. W. Guo, Jiang Zeng

备注：: 12 pages, see also http://math.univ-lyon1.fr/~guo
发表信息：: European J. Combin. 27 (2006), 884--895
DOI：: 10.1016/j.ejc.2005.04.009

英文摘要：

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-Salié 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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