On a Generalization of Carlitz’s Congruence
نویسندگان
چکیده
Let p be an odd prime and a be a positive integer. We show that p−1 k=0 (−1) (a−1)k p − 1 k a ≡ 2 a(p−1) + a(a − 1)(3a − 4) 48 p 3 B p−3 (mod p 4), which is a generalization of a congruence due to Carlitz.
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 کاملActa 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...
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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 ...
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