Abstract
In this paper, we give some identities of symmetry for the generalized degenerate Euler polynomials attached to \(\chi \) which are derived from the symmetric properties for certain fermionic p-adic integrals on \(\mathbb {Z}_{p}\).
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1 Introduction and Preliminaries
Let p be a fixed odd prime. Throughout this paper, \(\mathbb {Z}_{p},\mathbb {Q}_{p}\) and \(\mathbb {C}_{p}\) will be the ring of p-adic integers, the field of p-adic rational numbers and the completion of the algebraic closure of \(\mathbb {Q}_{p}\), respectively.
The p-adic norm \(\left| \cdot \right| _{p}\) in \(\mathbb {C}_{p}\) is normalized as \(\left| p\right| _{p}=\frac{1}{p}\). Let \(f\left( x\right) \) be continuous function on \(\mathbb {Z}_{p}\). Then the fermionic p-adic integral on \(\mathbb {Z}_{p}\) is defined as
From (1.1), we note that
where \(n\in \mathbb {N}\).
As is well known, the Euler polynomials are defined by the generating function
When \(x=0\), \(E_{n}=E_{n}\left( 0\right) \) are called the Euler numbers (see [1–19]).
For a fixed odd integer d with \(\left( p,d\right) =1\), we set
where \(a\in \mathbb {Z}\) lies in \(0\le a<dp^{N}\).
It is known that
where f is a continuous function on \(\mathbb {Z}_{p}\).
Let \(d\in \mathbb {N}\) with \(d\equiv 1\pmod {2}\) and let \(\chi \) be a Dirichlet character with conductor d. Then the generalized Euler polynomials attached to \(\chi \) are defined by the generating function
In particular, for \(x=0\), \(E_{n,\chi }=E_{n,\chi }\left( 0\right) \) are called the generalized Euler numbers attached to \(\chi \).
For \(d\in \mathbb {N}\) with \(d\equiv 1\pmod {2}\), by (1.2), we get
From (1.5), we have
Carlitz considered the degenerate Euler polynomials given by the generating function
Note that \(\lim _{\lambda \rightarrow 0}\mathcal {E}_{n}\left( x\mid \lambda \right) =E_{n}\left( x\right) \), \(\left( n\ge 0\right) \).
From (1.2), we note that
Thus, by (1.8), we get
where \(\left( x\mid \lambda \right) _{n}=x\left( x-\lambda \right) \cdots \left( x-\left( n-1\right) \lambda \right) \), for \(n\ge 1\), and \(\left( x\mid \lambda \right) _{0}=1\).
From (1.2), we can derive the following equation:
where \(d\in \mathbb {N}\) with \(d\equiv 1\pmod {2}\).
In view of (1.5), we define the generalized degenerate Euler polynomials attached to \(\chi \) as follows:
When \(x=0\), \(\mathcal {E}_{n,\lambda ,\chi }=\mathcal {E}_{n,\lambda ,\chi }\left( 0\right) \) are called the generalized degenerate Euler numbers attached to \(\chi \).
Let n be an odd natural number. Then, by (1.2), we get
Now, we set
From (1.2) and (1.12), we have
where \(n,d\in \mathbb {N}\) with \(n\equiv 1\pmod {2}\), \(d\equiv 1\pmod {2}\).
In this paper, we give some identities of symmetry for the generalized degenerate Euler polynomials attached to \(\chi \) derived from the symmetric properties of certain fermionic p-adic integrals on \(\mathbb {Z}_{p}\).
2 Identities of Symmetry for the Generalized Degenerate Euler Polynomials
Let \(w_{1},w_{2}\) be odd natural numbers. Then we consider the following integral equation:
From (1.10) and (1.11), we note that
By (1.14), we get
where \(k\ge 0\).
Thus, by (2.2) and (2.3), we get
where \(k\ge 0\), \(n,d\in \mathbb {N}\) with \(n\equiv 1\pmod {2}\), \(d\equiv 1\pmod {2}\).
Now, we set
From (2.5), we have
Thus, by (2.6), we see that \(I_{\chi }\left( w_{1},w_{2}\mid \lambda \right) \) is symmetric in \(w_{1},w_{2}\). By (1.12), (1.14), (2.2) and (2.5), we get
From the symmetric property of \(I_{\chi }\left( w_{1},w_{2}\mid \lambda \right) \) in \(w_{1}\) and \(w_{2}\), we have
Therefore, by (2.7) and (2.8), we obtain the following theorem.
Theorem 1
For \(w_{1},w_{2},d\in \mathbb {N}\) with \(w_{1}\equiv w_{2}\equiv d\equiv 1\pmod {2}\), let \(\chi \) be a Dirichlet character with conductor d. Then, we have
where \(l\ge 0\).
When \(x=0\), by Theorem 1, we get
By (2.5), we get
On the other hand,
Therefore, by (2.9) and (2.10), we obtain the following theorem.
Theorem 2
For \(w_{1},w_{2},d\in \mathbb {N}\) with \(d\equiv 1\pmod {2}\), \(w_{1}\equiv 1\pmod {2}\) and \(w_{2}\equiv 1\pmod {2}\), let \(\chi \) be a Dirichlet character with conductor d. Then, we have
To derive some interesting identities of symmetry for the generalized degenerate Euler polynomials attached to \(\chi \), we used the symmetric properties for certain fermionic p-adic integrals on \(\mathbb {Z}_{p}\). When \(w_{2}=1\), from Theorem 2, we have
In particular, for \(x=0\), we get
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Kim, D.S., Kim, T. (2016). Identities of Symmetry for the Generalized Degenerate Euler Polynomials. In: Singh, V., Srivastava, H., Venturino, E., Resch, M., Gupta, V. (eds) Modern Mathematical Methods and High Performance Computing in Science and Technology. Springer Proceedings in Mathematics & Statistics, vol 171. Springer, Singapore. https://doi.org/10.1007/978-981-10-1454-3_4
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