Abstract
In this paper, we consider Barnes-type special polynomials and give some identities of their polynomials which are derived from the bosonic p-adic integral or the fermionic p-adic integral on \(\mathbb{Z}_{p}\).
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1 Introduction
As is known, the Bernoulli polynomials of order r are defined by the generating function to be
When \(x=0\), \(B_{n}^{(r)}=B_{n}^{(r)}(0)\) are called the Bernoulli numbers of order r.
For \(a_{1},a_{2},\ldots,a_{r}\neq0\in\mathbb{C}_{p}\), the Barnes-Bernoulli polynomials are defined by the generating function to be
When \(x=0\), \(B_{n}(0|a_{1},a_{2},\ldots,a_{r})=B_{n}(a_{1},a_{2},\ldots,a_{r})\) are called Barnes Bernoulli numbers (see [14–33]).
Let p be a fixed odd prime number. Throughout this paper, \(\mathbb {Z}_{p}\), \(\mathbb{Q}_{p}\) and \(\mathbb{C}_{p}\) will denote the ring of p-adic integers, the field of p-adic numbers and the completion of algebraic closure of \(\mathbb{Q}_{p}\), respectively. The p-adic norm is defined as \(|p|_{p}=1/p\). Let \(UD(\mathbb{Z}_{p})\) be the space of uniformly differentiable function on \(\mathbb{Z}_{p}\). For \(f\in UD(\mathbb{Z}_{p})\), the bosonic p-adic integral on \(\mathbb{Z}_{p}\) is defined by
From (2), we have
where \(f_{1}(x)=f(x+1)\).
By using iterative method, we get
where \(f_{n}(x)=f(x+n)\) (\(n\in\mathbb{N}\)).
As is well known, the fermionic p-adic integral on \(\mathbb{Z}_{p}\) is defined by Kim to be
From (5), we can derive
In particular, \(n=1\), we have
The purpose of this paper is to investigate several special polynomials related to Barnes-type polynomials and give some identities including Witt’s formula of their polynomials.
Finally, we give some identities of mixed-type Bernoulli and Euler polynomials.
2 Barnes-type polynomials
Let \(a_{1},a_{2},\ldots,a_{r}\neq0\in\mathbb{C}_{p}\). Then, by (3), we get
From (8), we obtain the following Witt’s formula for the Barnes-Bernoulli polynomials.
Theorem 1
For \(a_{1},a_{2},\ldots,a_{r}\neq0\in\mathbb{C}_{p}\), we have
Note that
By (9) and Theorem 1, we obtain the following corollary.
Corollary 2
For \(n\geq2\), we have
where \(B_{n}=B_{n}(1)\) is the nth Bernoulli number.
From (2), we can easily derive the following integral equation:
where \(d\in\mathbb{N}\).
By (10), we get
Therefore, by (12), we obtain the following distribution relation for a Barnes-type Bernoulli polynomial.
Theorem 3
For \(n\geq0\), we have
From (4), we note that
By (13), we get
From (14), we can derive
By (15), we get
where \(n\in\mathbb{N}\) and \(m\in\mathbb{Z}\geq0\).
Therefore, by Theorem 1 and (16), we obtain the following theorem.
Theorem 4
For \(n\in\mathbb{N}\) and \(m\in\mathbb{Z}\) with \(m\geq0\), we have
Moreover,
From (15), we observe that
Thus, by (17), we get
where \(n\in\mathbb{N}\) and \(m\in\mathbb{Z}\geq0\).
Therefore, by Theorem 1 and (17), we obtain the following theorem.
Theorem 5
For \(n\in\mathbb{N}\) and \(m\geq0\), we have
Moreover,
Remark
Let \(a_{1}=1\) and \(r=1\). Then we have
Thus, we have
By (19), we get
where \(n\in\mathbb{N}\) and \(m\in\mathbb{Z}\geq0\).
It is easy to show that
where \(B_{n}(x)\) is the nth Bernoulli polynomial.
Thus, by (21), we get
From (20) and (21), we note that
where \(m\in\mathbb{Z}\geq0\) and \(n\in\mathbb{N}\).
From (5) and (6), we can derive the following equation:
Thus, we have
where \(E_{n}(x)\) is the nth Euler polynomial.
Witt’s formula for the Euler polynomials is given by
When \(x=0\), \(E_{n}=E_{n}(0)\) are called the Euler numbers.
For \(r\in\mathbb{N}\), the generating function of higher-order Euler polynomials can be derived from the multivariate p-adic fermionic integral on \(\mathbb{Z}_{p}\) as follows:
Thus we get
It is easy to show that
When \(x=0\), \(E^{(r)}_{n}=E^{(r)}_{n}(0)\) are called the higher-order Euler numbers.
From (6), we note that
Thus, by (27), we get
and
By comparing the coefficients on the both sides of (29), we get
where \(n\in\mathbb{N}\) and \(m\in\mathbb{Z}\geq0\).
Therefore, by (23) and (30), we obtain the following lemma.
Lemma 6
For \(m\geq0\), \(n\in\mathbb{N}\), we have
Let us assume that \(n\in\mathbb{N}\) with \(n\equiv1\ (\operatorname{mod}2)\).
Then, by (28), we get
Now, we consider the multivariate p-adic fermionic integral on \(\mathbb{Z}_{p}\) related to the higher-order Euler numbers as follows:
Thus, from (32), we can derive the following equation:
where \(m\geq0\) and \(n\in\mathbb{N}\) with \(n\equiv1\ (\operatorname{mod}2)\).
Therefore, by (24) and (33), we obtain the following theorem.
Theorem 7
For \(m\geq0\) and \(n\in\mathbb{N}\) with \(n\equiv1\ (\operatorname{mod}2)\), we have
Moreover,
From (32), we note that
where \(n\in\mathbb{N}\) with \(n\equiv1\ (\operatorname{mod}2)\).
Thus, by (34), we get
where \(m\in\mathbb{Z}\geq0\), \(n\in\mathbb{N}\) with \(n\equiv1 \ (\operatorname{mod}2)\).
Therefore by (24) and (35), we obtain the following theorem.
Theorem 8
For \(m\in\mathbb{Z}\geq0\), \(n\in\mathbb{N}\) with \(n\equiv1 \ (\operatorname{mod}2)\), we have
Moreover,
For \(a_{1}, a_{2},\ldots, a_{r}\in\mathbb{C}_{p}\backslash\{0\}\), let us consider the Barnes-type multiple Euler polynomials as follows:
When \(x=0\), \(E_{n}(a_{1},\ldots,a_{r})=E_{n}(0|a_{1},\ldots,a_{r})\) is called the nth Barnes-type Euler number.
For \(d\in\mathbb{N}\) with \(d\equiv1\ (\operatorname{mod}2)\), we observe that
From (37), we can derive the following equation:
By (38), we get
Therefore, by (36) and (39), we obtain the following theorem.
Theorem 9
For \(d\in\mathbb{N}\) with \(d\equiv1\ (\operatorname{mod}2)\), \(n\geq0\), we have
Moreover,
Remark
Note that
Thus, we have
Now, we define mixed-type Barnes-type Euler and Bernoulli numbers as follows:
where \(a_{1},\ldots,a_{r},b_{1},\ldots,b_{s}\neq0\).
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Lim, D., Do, Y. Some identities of Barnes-type special polynomials. Adv Differ Equ 2015, 42 (2015). https://doi.org/10.1186/s13662-015-0385-y
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DOI: https://doi.org/10.1186/s13662-015-0385-y