A 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 we expand on the left. This relation can also be written e (B+1)t − e Bt = t, or, if we equate power of t, B 0 = 1, (B + 1) k − B k = 1 if k = 1 0 if k > 1, where again we must first expand and then replace B i by B i , cf. [6,8,9,10,11]. Carlitz's q-Bernoulli numbers β k can be determined inductively by β 0 = 1, q(qβ + 1) k − β k = 1 if k = 1 0 if k > 1, (1) with the usual convention about replacing β i by β i,q (see [1,2,3,4,5,12]). Carlitz also defined q-Euler numbers and polynomials as H 0 (u; q) = 1, (qH + 1) k − uH k (u; q) = 0 for k ≥ 1, (2) where u is a complex number with |u| > 1; and for k ≥ 0,