- Research
- Open Access
On \((h,q)\)-Daehee numbers and polynomials
- Younghae Do1 and
- Dongkyu Lim1Email author
https://doi.org/10.1186/s13662-015-0445-3
© Do and Lim; licensee Springer. 2015
- Received: 10 February 2015
- Accepted: 17 March 2015
- Published: 31 March 2015
Abstract
The p-adic q-integral (sometimes called q-Volkenborn integration) was defined by Kim. From p-adic q-integral equations, we can derive various q-extensions of Bernoulli polynomials and numbers. DS Kim and T Kim studied Daehee polynomials and numbers and their applications. Kim et al. introduced the q-analogue of Daehee numbers and polynomials which are called q-Daehee numbers and polynomials. Lim considered the modified q-Daehee numbers and polynomials which are different from the q-Daehee numbers and polynomials of Kim et al. In this paper, we consider \((h,q)\)-Daehee numbers and polynomials and give some interesting identities. In case \(h=0\), we cover the q-analogue of Daehee numbers and polynomials of Kim et al. In case \(h=1\), we modify q-Daehee numbers and polynomials. We can find out various \((h,q)\)-related numbers and polynomials which are studied by many authors.
Keywords
- \((h,q)\)-Daehee numbers
- \((h,q)\)-Daehee polynomials
- \((h,q)\)-Bernoulli polynomials
- p-adic q-integral
MSC
- 11B68
- 11S40
1 Introduction
Let p be a fixed prime number. Throughout this paper, \(\mathbb{Z}_{p}\), \(\mathbb{Q}_{p}\) and \(\mathbb{C}_{p}\) will respectively denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completion of algebraic closure of \(\mathbb{Q}_{p}\). The p-adic norm is defined \(|p|_{p}=\frac{1}{p}\).
When \(x=0\), \(B^{(h)}_{n}(0|q)=B^{(h)}_{n}(q)\) are the nth \((h,q)\)-Bernoulli numbers.
The p-adic q-integral (or q-Volkenborn integration) was defined by Kim (see [1, 2]). From p-adic q-integral equations, we can derive various q-extensions of Bernoulli polynomials and numbers (see [1–24]). In [20], DS Kim and T Kim studied Daehee polynomials and numbers and their applications. In [3], Kim et al. introduced the q-analogue of Daehee numbers and polynomials which are called q-Daehee numbers and polynomials. Lim considered in [4] the modified q-Daehee numbers and polynomials which are different from the q-Daehee numbers and polynomials of Kim et al. In this paper, we consider \((h,q)\)-Daehee numbers and polynomials and give some interesting identities. In case \(h=0\), we cover the q-analogue of Daehee numbers and polynomials of Kim et al. (see [3]). In case \(h=1\), we have modified q-Daehee numbers and polynomials in [4]. We can find out various \((h,q)\)-related numbers and polynomials in [10, 13, 14].
2 \((h,q)\)-Daehee numbers and polynomials
From (14) and (15), if \(h=0\), \(D^{(0)}_{n}(q)\) is just the q-Daehee numbers which are defined by Kim et al. in [3]. If \(h=1\), \(D^{(1)}_{n}(q)\) is just the modified q-Daehee numbers which are studied in [4].
When \(x=0\), \(D^{(h)}_{n}(0|q)=D^{(h)}_{n}(q)\) is called the nth \((h,q)\)-Daehee number.
Thus we have the following theorem, which relates \((h,q)\)-Bernoulli polynomials and \((h,q)\)-Daehee polynomials.
Theorem 1
Therefore, we obtain the following theorem.
Theorem 2
Thus, we state the following theorem, which relates \((h,q)\)-Daehee numbers and \((h,q)\)-Bernoulli numbers.
Theorem 3
Thus, we have the following identity.
Theorem 4
Therefore, we get the following theorem, which relates \((h,q)\)-Daehee polynomials of the first and the second kind.
Theorem 5
Declarations
Acknowledgements
Authors wish to express their sincere gratitude to the referees for their valuable suggestions and comments.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
- Kim, T: q-Volkenborn integration. Russ. J. Math. Phys. 9(3), 288-299 (2002) MATHMathSciNetGoogle Scholar
- Kim, T: q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ. J. Math. Phys. 15(1), 51-57 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Kim, T, Lee, S-H, Mansour, T, Seo, J-J: A note on q-Daehee polynomials and numbers. Adv. Stud. Contemp. Math. 24(2), 155-160 (2014) MATHGoogle Scholar
- Lim, D: Modified q-Daehee numbers and polynomials. J. Comput. Anal. Appl. (2015, submitted) Google Scholar
- Kwon, J, Park, J-W, Pyo, S-S, Rim, S-H: A note on the modified q-Euler polynomials. JP J. Algebra Number Theory Appl. 31(2), 107-117 (2013) MATHMathSciNetGoogle Scholar
- Moon, E-J, Park, J-W, Rim, S-H: A note on the generalized q-Daehee numbers of higher order. Proc. Jangjeon Math. Soc. 17(4), 557-565 (2014) MATHMathSciNetGoogle Scholar
- Ozden, H, Cangul, IN, Simsek, Y: Remarks on q-Bernoulli numbers associated with Daehee numbers. Adv. Stud. Contemp. Math. (Kyungshang) 18(1), 41-48 (2009) MathSciNetGoogle Scholar
- Park, J-W: On the twisted Daehee polynomials with q-parameter. Adv. Differ. Equ. 2014, 304 (2014) View ArticleGoogle Scholar
- Park, J-W, Rim, S-H, Kwon, J: The twisted Daehee numbers and polynomials. Adv. Differ. Equ. 2014, 1 (2014) View ArticleMathSciNetGoogle Scholar
- Ryoo, CS: A note on the \((h,q)\)-Bernoulli polynomials. Far East J. Math. Sci. 41(1), 45-53 (2010) MATHMathSciNetGoogle Scholar
- Ryoo, CS, Kim, T: A new identities on the q-Bernoulli numbers and polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 21(2), 161-169 (2011) MATHMathSciNetGoogle Scholar
- Seo, JJ, Rim, S-H, Kim, T, Lee, SH: Sums products of generalized Daehee numbers. Proc. Jangjeon Math. Soc. 17(1), 1-9 (2014) MATHMathSciNetGoogle Scholar
- Simsek, Y: Twisted \((h,q)\)-Bernoulli numbers and polynomials related to twisted \((h,q)\)-zeta function and L-function. J. Math. Anal. Appl. 324(2), 790-804 (2006) View ArticleMATHMathSciNetGoogle Scholar
- Simsek, Y: The behavior of the twisted p-adic \((h,q)\)-L-functions at \(s=0\). J. Korean Math. Soc. 44(4), 915-929 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Simsek, Y, Rim, S-H, Jang, L-C, Kang, D-J, Seo, J-J: A note on q-Daehee sums. J. Anal. Comput. 1(2), 151-160 (2005) MathSciNetGoogle Scholar
- Srivastava, HM, Kim, T, Jang, L-C, Simsek, Y: q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series. Russ. J. Math. Phys. 12(2), 241-268 (2005) MATHMathSciNetGoogle Scholar
- Araci, S, Acikgoz, M, Esi, A: A note on the q-Dedekind-type Daehee-Changhee sums with weight α arising from modified q-Genocchi polynomials with weight α. J. Assam Acad. Math. 5, 47-54 (2012) MathSciNetGoogle Scholar
- Bayad, A: Modular properties of elliptic Bernoulli and Euler functions. Adv. Stud. Contemp. Math. (Kyungshang) 20(3), 389-401 (2010) MATHMathSciNetGoogle Scholar
- Dolgy, DV, Kim, T, Rim, S-H, Lee, SH: Symmetry identities for the generalized higher-order q-Bernoulli polynomials under \(S_{3}\) arising from p-adic Volkenborn integral on \(\Bbb{Z}_{p}\). Proc. Jangjeon Math. Soc. 17(4), 645-650 (2014) MATHMathSciNetGoogle Scholar
- Kim, DS, Kim, T: Daehee numbers and polynomials. Appl. Math. Sci. (Ruse) 7(120), 5969-5976 (2013) MathSciNetGoogle Scholar
- Kim, DS, Kim, T: q-Bernoulli polynomials and q-umbral calculus. Sci. China Math. 57(9), 1867-1874 (2014) View ArticleMATHMathSciNetGoogle Scholar
- Kim, DS, Kim, T, Komatsu, T, Lee, S-H: Barnes-type Daehee of the first kind and poly-Cauchy of the first kind mixed-type polynomials. Adv. Differ. Equ. 2014, 140 (2014) View ArticleMathSciNetGoogle Scholar
- Kim, DS, Kim, T, Lee, S-H, Seo, J-J: Higher-order Daehee numbers and polynomials. Int. J. Math. Anal. 8(6), 273-283 (2014) MathSciNetGoogle Scholar
- Kim, DS, Kim, T, Seo, J-J: Higher-order Daehee polynomials of the first kind with umbral calculus. Adv. Stud. Contemp. Math. (Kyungshang) 24(1), 5-18 (2014) MATHMathSciNetGoogle Scholar