On the q-Bernoulli Numbers and Polynomials with Weight α

and Applied Analysis 3 The purpose of this paper is to derive a new concept of higher-order q-Bernoulli numbers and polynomials with weight α from the fermionic p-adic q-integral on Zp. Finally, we present a systemic study of some families of higher-order q-Bernoulli numbers and polynomials with weight α. 2. Higher Order q-Bernoulli Numbers with Weight α Let β ∈ Z and α ∈ N in this paper. For k ∈ N and n ∈ Z , we consider the expansion of higher-order q-Bernoulli polynomials with weight α as follows:


Introduction
Let p be a fixed odd prime number.Throughout this paper, Z p , Q p , and C p will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of Q p , respectively.The p-adic norm of C p is defined as |x| p p −r , where x p r m/n with p, m p, n 1, r ∈ Q and m, n ∈ Z.Let N and Z be the set of natural numbers and integers, respectively, Z N ∪ {0}.Let q ∈ C p with |1 − q| p < p −1/ p−1 .The notation of q-number is defined by x w 1 − w x / 1 − w and x q 1 − q x / 1 − q , see 1-13 .
As the well known definition, the Bernoulli polynomials are defined by In the special case, x 0, B n 0 B n are called the nth Bernoulli numbers.That is, the recurrence formula for the Bernoulli numbers is given by with the usual convention about replacing B i with B i .In 1, 2 , q-extension of Bernoulli numbers are defined by Carlitz as follows: with the usual convention about replacing β i with β i,q .By 1.2 and 1.3 , we get lim q → 1 β i,q B i .In this paper, we assume that α ∈ N.
In 7 , the q-Bernoulli numbers with weight α are defined by Kim as follows: with the usual convention about replacing β α i with β α i,q .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 as 1.5 see 4, 5 .From 1.5 , we note that where f n x f x n and f l df x /dx | x l .
By 1.4 , 1.5 , and 1.6 , we set x n q α dμ q x , where n ∈ Z , 1.7 see 7 .The q-Bernoulli polynomials are also given by The purpose of this paper is to derive a new concept of higher-order q-Bernoulli numbers and polynomials with weight α from the fermionic p-adic q-integral on Z p .Finally, we present a systemic study of some families of higher-order q-Bernoulli numbers and polynomials with weight α.

Higher Order q-Bernoulli Numbers with Weight α
Let β ∈ Z and α ∈ N in this paper.For k ∈ N and n ∈ Z , we consider the expansion of higher-order q-Bernoulli polynomials with weight α as follows: From 2.1 , we note that Therefore, we obtain the following theorem.
Theorem 2.1.For n ∈ Z and k ∈ N, we have In the special case, x 0, β β,k|α n,q 0 β β,k|α n,q are called the nth higher order q-Bernoulli numbers with weight α.
From 2.1 and 2.2 , we can derive 2.4 By Theorem 2.1 and 2.4 , we get

2.5
From 2.1 , we have 2.6 Thus, we obtain the following theorem.
Theorem 2.2.For i ∈ N, we have It is easy to show that m j 0

Polynomials β 0,k|α n,q x
In this section, we consider the polynomials β 0,k|α n,q x as follows: From 3.1 , we can easily derive the following equation: By 3.1 and 3.2 , we get Therefore, by 3.3 and 3.4 , we obtain the following theorem.
Theorem 3.1.For n ∈ Z , we have Let d ∈ N.Then, we have

3.7
Thus, by 3.1 and 3.7 , we obtain the following theorem.

Polynomials β h,1|α n,q x
For h ∈ Z, let us define weighted h, q -Bernoulli polynomials β h,1|α n,q x as follows: 4.1 By 4.1 , we easily see that Therefore, by 4.2 , we obtain the following theorem.
Theorem 4.1.For h ∈ Z and n ∈ Z , we have From 4.1 , we can derive the following equation:

4.11
From 4.6 and 4.11 , we note that If we take x 0 in 4.12 , then we have

4.13
Therefore, by 4.12 and 4.13 , we obtain the following theorem.

Theorem 4.2. For h ∈ Z , we have
4.14 By 4.7 and Theorem 4.2, we obtain the following corollary.
From 4.1 , we have x .

4.17
Therefore, by 4.17 , we obtain the following theorem.

4.18
For x 1 in Theorem 4.4, we get

4.19
Therefore, by 4.19 , we obtain the following corollary.
Corollary 4.5.For h ∈ Z and n ∈ N with n > 1, we have Let d ∈ N. By 4.1 , we see that

4.21
By 4.1 and 4.21 , we obtain the following equation: where d ∈ N and h ∈ Z .

Polynomials β h,k|α n,q
x and h k From 2.1 , we note that From 5.1 , we have x q αx nα α q β h,k−1|α n,q x .

5.3
Therefore, by 5.3 , we obtain the following theorem.

5.4
It is easy to show that x .

5.5
Thus, by 5.5 , we obtain the following proposition.
Proposition 5.2.For h, n ∈ Z , we have x .5.6 For d ∈ N, we get

5.7
Thus, we obtain Equation 5.8 is multiplication formula for the q-Bernoulli polynomials of order h, k with weight α.
Let us define β k,k|α n,q x β k|α n,q x .Then we see that x .

5.10
Therefore, by 5.9 and 5.10 , we obtain the following theorem.
Theorem 5.3.For n ∈ Z and k ∈ N, we have x .

5.11
Let x k in Theorem 5.3.Then we see that

5.12
From Theorem 5.1, we can derive the following equation: x nq αx α α q β k,k−1|α n,q x .5.13 By 5.9 , we easily get From the definition of p-adic q-integral on Z p , we note that

5.15
Abstract and Applied Analysis 13 Thus, by 5.15 , we get n l 0 n l q α − 1 l β k|α n,q αn k k! αn k q k q ! .

5.16
By the definition of polynomial β k|α n,q x , we see that x n−l q q αx β k|α q x q α n , where n ∈ Z , 5.17 with the usual convention about replacing β k|α q n with β k|α n,q .Let x 0 in 5.13 .Then we have 5.18