In this paper, we consider the generalized Boole numbers and polynomials attached to a Dirichlet character \(\chi \) and investigate some properties ...
Abstract In this paper, we consider the generalized Boole numbers and polynomials attached to a Dirichlet character χ and investigate some properties of those numbers and polynomials. Keywords calculus
1 Introduction Let p be a fixed odd prime number and let Z p , Q p and C p denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completion of algebraic closure of Q p . The p-adic absolute value on C p is normalized so that | p| p = 1p . Let C Z p be the space of continuous functions on Z p . For f ∈ C Z p , the fermionic p-adic integral on Z p is defined by Kim to be I−1 ( f ) =
Zp
f (x) dμ−1 (x) = lim
n→∞
n −1 p
f (x) (−1)x ,
(1.1)
x=0
(see [15,20]). By (1.1), we get I−1 ( f 1 ) + I−1 ( f ) = 2 f (0), (see [15,20]), where f 1 (x) = f (x + 1).
B
Taekyun Kim tkkim@kw.ac.kr Dae San Kim dskim@sogang.ac.kr
1
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
2
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
(1.2)
D. S. Kim, T. Kim
From (1.2), we can derive the following equation : I−1 ( f n ) + (−1)n−1 I−1 ( f ) = 2
n−1
(−1)n−1−l f (l) ,
(1.3)
l=0
where f n (x) = f (x + n), n ∈ N. For d ∈ N with ( p, d) = 1, let X = X d = lim Z/d p N Z, X 1 = Z p , ← N