- Research
- Open Access
- Published:
Some explicit identities on Changhee-Genocchi polynomials and numbers
Advances in Difference Equations volume 2016, Article number: 202 (2016)
Abstract
In this paper, we introduce a new family of functions, which is called the Changhee-Genocchi polynomials. We study some explicit identities on these polynomials, which are related to Genocchi polynomials and Changhee polynomials. Also, we represent Changhee-Genocchi polynomials by gamma and beta functions.
We also study some properties of higher-order Changhee-Genocchi polynomials related to Changhee polynomials and Daehee polynomials.
1 Introduction
The Genocchi polynomials are defined by the generating function (see [1, 2])
When \(x=0\), \(G_{n}=G_{n}(0)\) are called the Genocchi numbers. From (1) we see that
We consider Changhee-Genocchi polynomials defined by the generating function
When \(x=0\), \(CG_{n} = CG_{n}(0)\) are called the Changhee-Genocchi numbers.
The gamma and beta functions are defined by the following definite integrals:
and
From (4) and (5) we have (see [3])
We recall that the classical Stirling numbers of the first kind \(S_{1}(n,k)\) and \(S_{2}(n,k)\) are defined by the relations (see [4])
respectively. Here \((x)_{n} = x(x-1)\cdots(x-n+1)\) denotes the falling factorial polynomial of order n. We also have
In this paper, we introduce a new family of functions, which is called the Changhee-Genocchi polynomials.
We study some properties of these polynomials, which are related to Genocchi polynomials and Changhee polynomials. Also we represent Changhee-Genocchi polynomials by gamma and beta functions.
We also study higher-order Changhee-Genocchi polynomials related to Changhee polynomials and Daehee polynomials.
Most of the ideas in this paper come from Kim and Kim [5]. Specifically, equations (14), (21), and (22) are related to the papers [5–8].
2 Changhee-Genocchi polynomials
First, we relate our newly defined Changhee-Genocchi polynomials to Genocchi polynomials.
Replacing t by \(e^{t}-1\) in (3) and applying (7), we get
The left-hand side of (8) is the generating function of the Genocchi polynomials.
Thus, by comparing the coefficients of (1) and (8) we have the following theorem.
Theorem 1
For any nonnegative integer k, we have
On the other hand, if we replace t by \(\log(1+t)\) in (1) and apply (7), then we get
where \(S_{1}(k,n)\) are the Stirling numbers of the first kind.
By comparing the coefficients of both sides of (10), we get the following theorem.
Theorem 2
For any nonnegative integer k, we have
Remark
When \(x=0\) in (11), we can see that Changhee-Genocchi numbers are integers.
We can consider equation (11) as the inversion formula for (9). From (3) we can consider the following identity:
Thus, by comparing the coefficients of both sides of (12) we have
From (13) we can derive the following theorem.
Theorem 3
For any nonnegative integer n, we have
In this paper, we define the λ-Changhee-Genocchi polynomials by a generating function as follows:
We recall that the λ-Changhee polynomials are defined in [9] by
When \(\lambda=1\), Changhee-Genocchi polynomials are well-known Changhee polynomials, cf. [10–18]. In order to establish a reflexive symmetry on the Changhee-Genocchi polynomials, we consider the following:
By comparing the coefficients of (17) we have the following theorem.
Theorem 4
For \(n\in\mathbb {N}\), we have
Thus, from (3) and (18) we have
By comparing the coefficients of (19) we have
On the other hand, by (5), (6), and (20) we have
Thus, by (18) and (21) we have the following identities, which relate the λ-Changhee-Genocchi polynomials, the Stirling numbers, and the beta and gamma polynomials:
From (16) we consider
By comparing the coefficients of (23) we have the following theorem.
Theorem 5
For any nonnegative integer n, we have
Remark
If we take \(\lambda=1\) in Theorem 5, then we have the result in Theorem 4.
From the second line of (23) and from (16) we have
By comparing the coefficients of (23) and (25) we have the following theorem.
Theorem 6
For any positive integer n, we have
For \(r\in\mathbb {N}\), we define the Changhee-Genocchi polynomials \(CG_{n}^{(r)}(x)\) of order r by the generating function
From (26) we have the following relation between the Changhee-Genocchi polynomials of order r and the Changhee polynomials of order r:
By comparing the coefficients of (26) and (27) we have the following theorem.
Theorem 7
For any nonnegative integer n, we have
For \(d\in\mathbb {N}\) with \(d\equiv1\ (\operatorname{mod}2)\), we have the following identity:
So, for such \(d\equiv1\ (\operatorname{mod} 2)\), from (28), (3), and (15) we see that
By comparing the coefficients in (29), for \(d\equiv1\ (\operatorname{mod} 2)\), we have the following theorem.
Theorem 8
For any nonnegative integer n and \(d\equiv1\ (\operatorname{mod} 2)\), we have
We remark that, for \(d\equiv1\ (\operatorname{mod} 2)\), from (9) and (30) we have the inversion of Theorem 8.
Theorem 9
For any nonnegative integer n and \(d\equiv1\ (\operatorname{mod} 2)\), we have
From the generating function of the Changhee-Genocchi polynomials in (1), replacing t by \(\lambda\log(1+t)\), we get
Thus, the left-hand side of (31) can be represented by the λ-Changhee-Genocchi polynomials as follows:
By comparing the coefficients of (31) and (32) we have the following theorem.
Theorem 10
For any nonnegative integer k, we have
From the generating function of the Changhee-Genocchi numbers in (3) we want to see the recurrence relation for the Changhee-Genocchi numbers:
On the other hand, from the left-hand side of (33) we have
By comparing the coefficients of (33) and (34) we have the following recurrence relation for the Changhee-Genocchi numbers.
Theorem 11
We have
From the higher-order Changhee-Genocchi polynomials
we can deduce
Thus, from (36) we can rewrite (35) as follows:
We recall that the Dahee polynomials are defined by the generating function (see [9, 19])
When \(x=0\), \(D_{n} = D_{n}(0)\) are called the Dahee numbers.
For \(r\in\mathbb {N}\), the higher-order Daehee numbers are given by the generating function (see [9, 19, 20])
From (28) we have
Thus, from (38) we have the following theorem.
Theorem 12
For any nonnegative integer n and \(d\equiv1\ (\operatorname{mod} 2)\), we have
3 Changhee-Genocchi polynomials arising from differential equations
In this section, we give new identities on the Changhee-Genocchi numbers by using differential equations. We use the idea recently developed by Kwon et al. [21].
By equation (3) we can write the generating function for the Changhee-Genocchi numbers as follows:
Let
Then
Thus,
On the other hand,
From (39) we have
By comparing the coefficients of (40), (41), and (42) we have new identities on the Changhee-Genocchi numbers as follows.
Theorem 13
For any nonnegative integer s, we have
References
Kim, T: On the multiple q-Genocchi and Euler numbers. Russ. J. Math. Phys. 15(4), 481-486 (2008)
Srivastava, HM, Özarslan, MA, Kaanoğlu, C: Some generalized Lagrange-based Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials. Russ. J. Math. Phys. 20(1), 110-120 (2013)
Zill, DG, Cullen, MR: Advanced Engineering Mathematics. Jones & Bartlett, Sudbury (2006)
Roman, S: The Umbral Calculus. Pure and Applied Mathematics, vol. 111, x+193 pp. Academic Press [Harcourt Brace Jovanovich, Publishers], New York (1984)
Kim, DS, Kim, T: Some identities involving Genocchi polynomials and numbers. Ars Comb. 121, 403-412 (2015)
Kim, DS, Kim, T: A study on the integral of the product of several Bernoulli polynomials. Rocky Mt. J. Math. 44(4), 1251-1263 (2014)
Kim, T: Some properties on the integral of the product of several Euler polynomials. Quaest. Math. 38, 553-562 (2015)
Kim, T: A study on the q-Euler numbers and the fermionic q-integral of the product of several type q-Bernstein polynomials on \(\mathbb {Z}_{p}\). Adv. Stud. Contemp. Math. (Kyungshang) 23, 5-11 (2013)
Kwon, HI, Kim, T, Seo, JJ: A note on degenerate Changhee numbers and polynomials. Proc. Jangjeon Math. Soc. 18(3), 295-305 (2015)
Ozden, H, Cangul, IN, Simsek, Y, Kurt, V: On the higher-order w-q-Genocchi numbers. Adv. Stud. Contemp. Math. 19(1), 39-57 (2009)
Jang, L-C, Ryoo, CS, Seo, JJ, Kwon, HI: Some properties of the twisted Changhee polynomials and their zeros. Appl. Math. Comput. 274, 169-177 (2016)
Kim, T: p-Adic q-integrals associated with the Changhee-Barnes’ q-Bernoulli polynomials. Integral Transforms Spec. Funct. 15(5), 415-420 (2004)
Kim, T, Dolgy, DV, Kim, DS, Seo, JJ: Differential equations for Changhee polynomials and their applications. J. Nonlinear Sci. Appl. 9, 2857-2864 (2016)
Kim, DS, Kim, T: A note on Changhee polynomials and numbers. Adv. Stud. Theor. Phys. 7(20), 993-1003 (2013)
Kim, T, Kim, DS: A note on nonlinear Changhee differential equations. Russ. J. Math. Phys. 23(1), 88-92 (2016)
Kim, DS, Kim, T, Seo, JJ, Lee, SH: Higher-order Changhee numbers and polynomials. Adv. Stud. Theor. Phys. 8(8), 365-373 (2014)
Kim, T, Rim, S-H: New Changhee q-Euler numbers and polynomials associated with p-adic q-integrals. Comput. Math. Appl. 54(4), 484-489 (2007)
Rim, S-H, Pak, HK, Jeong, J, Kang, DJ: Changhee-Genocchi numbers and their applications. Submitted for publication
El-Desouky, BS, Mustafa, A: New results on higher-order Daehee and Bernoulli numbers and polynomials. Adv. Differ. Equ. 2016, 32 (2016)
Wang, NL, Li, H: Some identities on the higher-order Daehee and Changhee numbers. Pure Appl. Math. 4(5-1), 33-37 (2015)
Kwon, HI, Kim, T, Seo, JJ: A note on Daehee numbers arising from differential equations. Glob. J. Pure Appl. Math. 12(3), 2349-2354 (2016)
Acknowledgements
The authors would like to express their sincere gratitude to the Editor, who gave us valuable comments to improve this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kim, BM., Jeong, J. & Rim, SH. Some explicit identities on Changhee-Genocchi polynomials and numbers. Adv Differ Equ 2016, 202 (2016). https://doi.org/10.1186/s13662-016-0925-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-016-0925-0
MSC
- 05A10
- 05A19
- 11B68
- 11S80
Keywords
- Euler polynomials
- Changhee polynomials
- Genocchi polynomials
- Changhee-Genocchi numbers
- beta and gamma functions