Some Arithmetic Properties of the q-Euler Numbers and q-Sali\'e Numbers
Combinatorics
2011-03-25 v2
Abstract
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.
Cite
@article{arxiv.math/0504569,
title = {Some Arithmetic Properties of the q-Euler Numbers and q-Sali\'e Numbers},
author = {Victor J. W. Guo and Jiang Zeng},
journal= {arXiv preprint arXiv:math/0504569},
year = {2011}
}
Comments
12 pages, see also http://math.univ-lyon1.fr/~guo