q-Analogue of Riemann's ζ-function and q-Euler numbers

ON q - EULER NUMBERS AND q - SALIÉ

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

q-EULER NUMBERS AND POLYNOMIALS ASSOCIATED WITH p-ADIC q-INTEGRALS AND BASIC q-ZETA FUNCTION

The purpose of this paper is to introduce the some interesting properties of q-Euler numbers and polynomials, cf. [1, 2, 5]. Finally, we will consider the “ sums of products of q-Euler polynomials”.

q-EULER AND GENOCCHI NUMBERS

Carlitz has introduced an interesting q-analogue of Frobenius-Euler numbers in [4]. He has indicated a corresponding Stadudt-Clausen theorem and also some interesting congruence properties of the q-Euler numbers. In this paper we give another construction of q-Euler numbers, which are different than his q-Euler numbers. By using our q-Euler numbers, we define the q-analogue of Genocchi numbers ...

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...