MathematicsComputer Science

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

Victor J. W. Guo, Jiang Zeng

2005.4.28EUROPEAN JOURNAL OF COMBINATORICS

DOI: 10.1016/j.ejc.2005.04.009

tlooto Summary

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.

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-Salie 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.

Citation format

GUO, Victor J. W.; ZENG, Jiang. Some arithmetic properties of the q-euler numbers and q-salié numbers [preprint]. arXiv, 2005. arXiv:math/0504569.