Some Arithmetic Properties of the q-Euler Numbers and q-Salié Numbers
arXiv:math/0504569 · doi:10.1016/j.ejc.2005.04.009
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-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.
12 pages, see also http://math.univ-lyon1.fr/~guo
References in corpus (1)
Cited by in corpus (11)
- On Orbits of Order Ideals of Minuscule Posets
- Factors of binomial sums from the Catalan triangle
- A rooted-trees q-series lifting a one-parameter family of Lie idempotents
- On q-Euler numbers, q-Salie numbers and q-Carlitz numbers
- A q-analogue of Lehmer's congruence
- New congruences for sums involving Apery numbers or central Delannoy numbers
- Factors of sums involving -binomial coefficients and powers of -integers
- Proof of a conjecture involving Sun polynomials
- Factors of sums and alternating sums of products of -binomial coefficients and powers of -integers
- On the least common multiple of -binomial coefficients
- Some congruences related to the q-Fermat quotients