Acta Arith. 124(2006), no. 1, 41–57. ON q-EULER NUMBERS, q-SALIÉ NUMBERS AND q-CARLITZ NUMBERS
نویسنده
چکیده
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 of Guo and Zeng.
منابع مشابه
Arith . , in press . ON q - EULER NUMBERS , q - SALIÉ NUMBERS AND q -
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. Our result on q-Salié numbers implies a conjecture o...
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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 ...
متن کاملSome 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.
متن کامل2 Hao Pan and Zhi
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 ...
متن کاملA NOTE ON q-EULER NUMBERS AND POLYNOMIALS
The purpose of this paper is to construct q-Euler numbers and polynomials by using p-adic q-integral equations on Z p. Finally, we will give some interesting formula related to these q-Euler numbers and polynomials. The usual Bernoulli numbers are defined by ∞ k=0 B k t k k! = t e t − 1 , which can be written symbolically as e Bt = t e t −1 , interpreted to mean B k must be replaced by B k when...
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تاریخ انتشار 2006