Some arithmetic properties of the q-Euler numbers and q-Salié numbers

Authors:
Victor J. W. Guo;Jiang Zeng
Affiliations:
Institut Camille Jordan, Université Claude Bernard, Villeurbanne Cedex, France;Institut Camille Jordan, Université Claude Bernard, Villeurbanne Cedex, France
Venue:
European Journal of Combinatorics
Year:
2006

Citing 1
Cited 2

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Combinatorial Enumeration

On the divisibility of q-Salié numbers

Computers & Mathematics with Applications
On orbits of order ideals of minuscule posets

Journal of Algebraic Combinatorics: An International Journal

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Abstract

For m n ≥ 0 and 1 ≤ d ≤ m, it is shown that the q-Euler number E2m(q) is congruent to qm-n E2n(q) mod (1 + qd) if and only if m ≡ n mod d. The q-Salié number S2n(q) is shown to be divisible by (1 + q2r+1)⌊n/2r+1⌋ for any r ≥ 0. Furthermore, similar congruences for the generalized q-Euler numbers are also obtained, and some conjectures are formulated.