Abstract
In this paper, we consider Korobov-type polynomials derived from the bosonic and fermionic p-adic integrals on \(\mathbb{Z}_{p}\), and we give some interesting and new identities of those polynomials and of their mixed-types.
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1 Introduction
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 rational numbers and the completion of the algebraic closure of \(\mathbb{Q}_{p}\). Let \(\nu_{p}\) be the normalized exponential valuation of \(\mathbb{C}_{p}\) with \(\vert p\vert _{p}=p^{-\nu_{p} (p )}=\frac{1}{p}\). Let \(UD (\mathbb {Z}_{p})\) be the space of uniformly differentiable functions on \(\mathbb {Z}_{p}\). For \(f\in UD (\mathbb {Z}_{p})\), the bosonic p-adic integral on \(\mathbb {Z}_{p}\) is defined by
Thus, by (1.1), we get
The fermionic p-adic integral on \(\mathbb {Z}_{p}\) is defined by Kim as
Thus, from (1.3), we have
From (1.2) and (1.4), we can derive the following equations:
where \(f_{n} (x )=f (x+n )\), \(f^{\prime} (l )=\frac{df (x )}{dx}\vert _{x=l}\) (see [1, 4, 5]).
As is well known, the Bernoulli polynomials of order r (\(\in\mathbb {N}\)) are defined by the generating function
When \(x=0\), \(B_{n}^{ (r )}=B_{n}^{ (r )} (0 )\) are called the Bernoulli numbers of order r. In particular, if \(r=1\), \(B_{n} (x )=B_{n}^{ (1 )} (x )\) are called the ordinary Bernoulli polynomials.
The Euler polynomials of order r are also given by the generating function
When \(x=0\), \(E_{n}^{ (r )}=E_{n}^{ (r )} (0 )\) are called the Euler numbers of order r. In particular, if \(r=1\), then \(E_{n} (x )=E_{n}^{ (1 )} (x )\) are called the ordinary Euler polynomials.
The Daehee polynomials of order r are defined by the generating function
When \(x=0\), \(D_{n}^{ (r )}=D_{n}^{ (r )} (0 )\) are called the Daehee numbers of order r. In particular, if \(r=1\), then \(D_{n} (x )=D_{n}^{ (1 )} (x )\) are called the ordinary Daehee polynomials. Now, we introduce the Changhee polynomials of order r given by the generating function
When \(x=0\), \(\mathit {Ch}_{n}^{ (r )}=\mathit {Ch}_{n}^{ (r )} (0 )\) are called the Changhee numbers of order r. In particular, if \(r=1\), then \(\mathit {Ch}_{n} (x )=\mathit {Ch}_{n}^{ (1 )} (x )\) are called the ordinary Changhee polynomials.
Recently, Korobov introduced the special polynomials given by the generating function
Note that \(\lim_{\lambda\rightarrow0}K_{n} (x\mid\lambda )=b_{n} (x )\), where \(b_{n} (x )\) are the Bernoulli polynomials of the second kind defined by the generating function
In this paper, we define the higher-order Korobov polynomials given by the generating function
When \(x=0\), \(K_{n}^{ (r )} (\lambda )=K_{n}^{ (r )} (0\mid\lambda )\) are called the Korobov numbers of order r. In particular, if \(r=1\), then \(K_{n} (\lambda )=K_{n}^{ (1 )} (0\mid\lambda )=K_{n} (0\mid\lambda )\) are called the ordinary Korobov numbers. Now, we consider the Korobov-type Changhee polynomials which are called the λ-Changhee polynomials as follows:
When \(x=0\), \(\mathit {Ch}_{n} (\lambda )=\mathit {Ch}_{n} (0\mid\lambda )\) are called λ-Changhee numbers. Note that \(\lim_{\lambda \rightarrow1}\mathit {Ch}_{n} (x\mid\lambda )=\mathit {Ch}_{n} (x )\), \(\lim_{\lambda\rightarrow0}\mathit {Ch}_{n} (x\mid\lambda )= (x )_{n}\), where
For \(r\in\mathbb{N}\), the λ-Changhee polynomials of order r are defined by the generating function
The Stirling numbers of the second kind are defined by the generating function
The Korobov polynomials (of the first kind) were introduced in [10] as the degenerate version of the Bernoulli polynomials of the second kind. In recent years, many researchers studied various kinds of degenerate versions of some familiar polynomials like Bernoulli polynomials, Euler polynomials and their variants by means of generating functions, p-adic integrals and umbral calculus (see [1, 6, 12, 14]).
Here in this paper we introduce two Korobov-type polynomials obtained from the same function, namely the one by performing bosonic p-adic integrals on \(\mathbb{Z}_{p}\) and the other by carrying out fermionic p-adic integrals on \(\mathbb{Z}_{p}\). In addition, we consider their higher-order versions and some mixed-types of them by considering multivariate p-adic integrals. In conclusion, we will obtain some connections between these new polynomials and Bernoulli polynomials, Euler polynomials, Daehee numbers and Bernoulli numbers of the second kind.
2 Korobov-type polynomials
For \(\lambda\in\mathbb{N}\), by (1.2), we get
From (2.1), we have
Therefore, by (2.2), we obtain the following theorem.
Theorem 2.1
For \(n\ge0\), we have
Now, we observe that
Therefore by (2.3), we obtain the following corollary.
Corollary 2.2
For \(n\ge0\), we have
From (2.1), we have
Therefore, by (2.4), we obtain the following corollary.
Corollary 2.3
For \(n\ge0\), we have
By replacing t by \(e^{t}-1\) in (1.10), we get
On the other hand,
Therefore, by (2.5) and (2.6), we obtain the following theorem.
Theorem 2.4
For \(n\ge0\), we have
It is easy to show that
By (2.7), we get
On the other hand,
Thus, by (2.8) and (2.9), we get
Therefore, by (1.10) and (2.10), we obtain the following theorem.
Theorem 2.5
For \(n\ge0\) and \(d\in\mathbb{N}\), we have
From (1.5), we can derive the following equation:
Thus, by (2.11), we get
From (2.12), we have
On the other hand,
Therefore, by (2.13) and (2.14), we obtain the following theorem.
Theorem 2.6
For \(n\in\mathbb{N}\), \(m\ge0\), we have
Remark
By (1.5), we easily get
Hence, by Theorem 2.1 and (2.15), we see
Now, we consider the multivariate p-adic integral on \(\mathbb {Z}_{p}\) given by
Thus, by (2.17), we get
By comparing the coefficients on both sides, we obtain the following theorem.
Theorem 2.7
For \(n\ge0\), we have
By replacing t by \(e^{t}-1\) in (1.12), we get
On the other hand,
Therefore, by (2.18) and (2.19), we obtain the following theorem.
Theorem 2.8
For \(n\ge0\), we have
From (1.3), we can derive the following equation:
Thus, by (2.20), we get
We observe that
Therefore, by (2.21) and (2.22), we obtain the following theorem.
Theorem 2.9
For \(n\ge0\), we have
From (2.20), we have
By replacing t by \(e^{t}-1\) in (1.13), we get
On the other hand,
Therefore, by (2.23), (2.24), and (2.25), we obtain the following theorem.
Theorem 2.10
For \(n\ge0\), we have
and
By replacing t by \(e^{t}-1\) in (1.14), we get
From (1.14), we can derive the following equation:
Therefore, by (2.26) and (2.27), we obtain the following theorem.
Theorem 2.11
For \(n\ge0\), we have
and
Let us observe the following multivariate fermionic p-adic integral on \(\mathbb {Z}_{p}\):
Thus, by (2.28), we get
Note that
Thus, we get
By (2.28) and (2.29), we easily get
Now, we consider the λ-Changhee and Korobov mixed-type polynomials which are given by the multivariate p-adic integral on \(\mathbb {Z}_{p}\) as follows:
where \(r,s\in\mathbb{N}\).
Now, we observe that
By (2.33), we get
From (2.32) and (2.34), we have
The generating function of \(\mathit {CK}_{n}^{ (r,s )} (x\mid\lambda )\) is given by
Theorem 2.12
For \(r,s\in\mathbb{N}\) and \(n\ge0\), we have
We consider the Korobov and λ-Changhee mixed-type polynomials, which are given by
where \(r,s\in\mathbb{N}\) and \(n\ge0\).
Then, by (2.37), we get
The generating function of \(\mathit {KC}_{n}^{ (r,s )} (x\mid\lambda )\) is given by
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Acknowledgements
The authors would like to thank the referees and editor for their valuable comments. The work reported in this paper was conducted during the sabbatical year of Kwangwoon University in 2014.
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Kim, D.S., Kim, T. Some identities of Korobov-type polynomials associated with p-adic integrals on \(\mathbb{Z}_{p}\) . Adv Differ Equ 2015, 282 (2015). https://doi.org/10.1186/s13662-015-0602-8
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DOI: https://doi.org/10.1186/s13662-015-0602-8