Arith . , in press . ON q - EULER NUMBERS , q - SALIÉ NUMBERS AND q -

Acta Arith. 124(2006), no. 1, 41–57. ON q-EULER NUMBERS, q-SALIÉ NUMBERS AND q-CARLITZ NUMBERS

for any nonnegative integers n, s, t with 2 ∤ t, where [k]q = (1−q)/(1−q); this is a q-analogue of Stern’s congruence E2n+2s ≡ E2n +2 (mod 2s+1). We also prove that (−q; q)n = ∏ 0<k6n(1 + q ) divides S2n(q) and the numerator of C2n(q); this extends Carlitz’s result that 2 divides the Salié number S2n and the numerator of the Carlitz number C2n. Our result on q-Salié numbers implies a conjecture...

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

For m > n ≥ 0 and 1 ≤ d ≤ 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 ≡ n mod d. The q-Salié number S 2n (q) is shown to be divisible by (1 + q 2r+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.

Preprint (2005-05-25), arXiv:math.CO/0505548. ON q-EULER NUMBERS, q-SALIÉ NUMBERS AND q-CARLITZ NUMBERS

for any nonnegative integers n, s, t with 2 ∤ t, where [k]q = (1−q)/(1−q), this is a q-analogue of Stern’s congruence E2n+2s ≡ E2n +2 (mod 2s+1). We also prove that (−q; q)n = ∏ 0<k6n(1 + q ) divides S2n(q) and the numerate of C2n(q), this extends Carlitz’s result that 2 divides the Salié number S2n and the numerate of the Carlitz number C2n. For q-Salié numbers we also confirm a conjecture of ...

. CO / 0505548 . ON q - EULER NUMBERS AND q - SALIÉ NUMBERS

ON q - EULER NUMBERS AND q - SALIÉ