A note on the generalized q-Bernoulli measures with weight a

In this paper we consider the generalized q-Bernoulli measures with weight alpha. From those measures, we derive some interesting properties on the generalized q-Bernoulli numbers with weight alpha attached to chi.

Thus, we note that lim q→1 [x] q = x.
In [2], Carlitz defined a set of numbers ξ k =ξ k (q) inductively by with the usual convention of replacing ξ k by ξ k . These numbers are q-extension of ordinary Bernoulli numbers B k . But they do not remain finite when q = 1. So he modified (1) as follows: with the usual convention of replacing β k by β k,q . The numbers β k,q are called the k-th Carlitz q-Bernoulli numbers.
In [1], Carlitz also considered the extended Carlitz's q-Bernoulli numbers as follows: with the usual convention of replacing (β h ) k by β h k,q . Recently, Kim considered q-Bernoulli numbers, which are different extended Carlitz's q-Bernoulli numbers, as follows: for α ∈ N and n ∈ Z + , with the usual convention of replacing ( β (α) ) k by β (α) k,q (see [3]).
The numbers β (α) k,q are called the k-th q-Bernoulli numbers with weight α. For fixed d ∈ Z + with (p, d) = 1, we set where a ∈ Z satisfies the condition 0 ≤ a < dp N . Let UD(Z p ) be the space of uniformly differentiable functions on Z p . For f ∈ UD(Z p ), the p-adic q-integral on Z p is defined by Kim as follows: [3,4]).
By (3) and (4), the Witt's formula for the q-Bernoulli numbers with weight α is given by The q-Bernoulli polynomials with weight α are also defined by From (4), (5) and (6), we can derive the Witt's formula for β (α) n,q (x) as follows: For n ∈ Z + and d ∈ N, the distribution relation for the q-Bernoulli polynomials with weight α are known that Recently, several authors have studied the p-adic q-Euler and Bernoulli measures on Z p (see [8,9,11]). The purpose of this paper is to construct p-adic q-Bernoulli distribution with weight α(= p-adic q-Bernoulli unbounded measure with weight α) on Z p and to study their integral representations. Finally, we construct the generalized q-Bernoulli numbers with weight α and investigate their properties related to p-adic q-L-functions.

p-adic q-Bernoulli distribution with weight α
Let X be any compact-open subset of Q p , such as Z p or Z * p . A p-adic distribution µ on X is defined to be an additive map from the collection of compact open set in X to Q p : The set Z p has a topological basis of compact open sets of the form a + p n Z p .
Consequently, if U is any compact open subset of Z p , it can be written as a finite disjoint union of sets where N ∈ Z + and a 1 , a 2 , · · · , a k ∈ Z with 0 ≤ a i < p N − 1 Indeed, the p-adic ball a+p n Z p can be represented as the union of smaller balls

Lemma 1. Every map µ from the collection of compact-open sets in
holds whenever a + p N Z p ⊂ X, extends to a p-adic distribution on X. Now we define a map µ (α) k,q on the balls in Z p as follows: where {a} n is the unique number in the set {0, 1, 2, · · · , p n − 1} such that {a} n ≡ a (mod p n ). If a ∈ {0, 1, 2, · · · , p n − 1}, then From (10), we note that µ Theorem 2. Let α ∈ N and k ∈ Z + . Then we see that µ (α) k,q (a + p n Z p ) is p-adic distribution on Z p if and only if We set f From (9) and (11), we get By (8), (12) and Theorem 2, we obtain the following theorem.
k,q be given by k,q dp N a dp N .
Then µ (α) k,q extends to a Q(q)-valued distribution on the compact open sets U ⊂ X.
From (13), we note that k,q dp N a dp N .
Theorem 4. For α ∈ N and k ∈ Z + , we have Let χ be Dirichlet character with conductor d ∈ N. Then we define the generalized q-Bernoulli numbers attached to χ as follows: From (13) and (15), we can derive the following equation.
Let us define χ x χ y = χ x,k,α:q • χ y,k,α:q . Then we have From the definition of χ x , we can easily derive the following equation.