Abstract
We consider the modified degenerate q-Daehee polynomials and numbers of the second kind which can be represented as the p-adic q-integral. Furthermore, we investigate some properties of those polynomials and numbers.
Similar content being viewed by others
1 Introduction
Throughout this paper, \(\mathbb{Z}\), \(\mathbb{Q}\), \({\mathbb{Z}}_{p}\), \({\mathbb{Q}}_{p}\) and \({\mathbb{C}}_{p}\) will, respectively, denote the ring of integers, the field of rational numbers, the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of \({\mathbb{Q}}_{p}\). The p-adic norm \(\vert \cdot \vert _{p}\) is normalized by \(\vert p \vert _{p}=\frac{1}{p}\). If \(q \in {\mathbb{C}}_{p}\), we normally assume \(\vert q-1 \vert _{p}< p^{-\frac{1}{p-1}}\), so that \(q^{x} = \exp (x \log q)\) for \(\vert x \vert _{p} \le 1\). The q-extension of x is defined as \([x]_{q}=\frac{1-q^{x}}{1-q}\) for \(q\neq 1\) and x for \(q=1\) (see [3,4,5,6, 12, 17, 18, 20, 21, 25, 27, 29,30,31, 33,34,35, 41, 45, 46]). Let \(\operatorname{UD}(\mathbb{Z}_{p})\) be the space of uniformly differentiable functions on \(\mathbb{Z}_{p}\). For \(f \in \operatorname{UD}(\mathbb{Z}_{p} )\), Volkenborn integral (or p-adic bosonic integral) on \(\mathbb{Z}_{p}\) is given by
where \(\mu _{1}(x)=\mu _{1}(x+p^{N} {\mathbb{Z}}_{p}) \) denotes the Haar distribution defined by \(\mu _{1}(x+p^{N} {\mathbb{Z}}_{p})=\frac{1}{p ^{N}}\) (see [1, 2, 8,9,10,11,12,13,14, 16, 19, 24, 32, 35, 37,38,39,40,41,42,43,44, 46, 47]). Then, by (1.1), we get \(I(f_{1} ) -I_{1} (f) =f^{\prime } (0)\), where \(f_{1} (x) =f(x+1)\) and \(\frac{d}{dx} f(x)| _{x=0} =f^{\prime } (0)\).
For \(f \in \operatorname{UD}(\mathbb{Z}_{p})\), the p-adic q-integral on \(\mathbb{Z}_{p}\) is defined by Kim to be
(see [12, 17,18,19,20, 25, 29, 31, 33, 34, 47]). Note that
(see [6, 9, 18, 19, 21, 25, 28, 29, 32,33,34, 36, 38, 42, 43, 47]). Let \(f_{1} (x) =f(x+1)\). Then, by (1.2), we get
where \(f^{\prime } (0) = \frac{d}{dx} f(x)| _{x=0}\) (see [6, 9, 18, 19, 21, 25, 28, 29, 32,33,34, 36, 38, 42, 43, 47]).
Carlitz considered q-Bernoulli numbers which are recursively given by
with the usual convention about replacing \(\beta _{q}^{n}\) by \(\beta _{n,q}\) (see [3,4,5]). He also defined q-Bernoulli polynomials as
(see [3,4,5]). In [19], Kim proved that the Carlitz q-Bernoulli polynomials are represented by p-adic q-integral on \(\mathbb{Z} _{p}\) as follows:
In [17], Kim considered the modified q-Bernoulli polynomials which are different from Carlitz to be
When \(x=0\), \(B_{n,q}=B_{n,q}(0)\) are called the modified q-Bernoulli numbers (see [17, 18]). Thus, we note that
with the usual convention about replacing \(B_{q}^{n}\) by \(B_{n,q}\) (see [17, 18, 21, 25, 34]).
In [33, 35, 46], the authors studied the q-Daehee polynomials which are defined by the generating function to be
In [12], the authors studied the degenerate λ-q-Daehee polynomials as follows:
Like this idea of the Carlitz q-Bernoulli polynomials (1.4), we will define the modified q-Daehee polynomials of the second kind which are different from the modified q-Daehee numbers and polynomials in [31].
As is well known, the Stirling number of the first kind is defined by
and the Stirling number of the second kind is given by the generating function,
We also have
and
(see [7, 14, 15, 22, 23, 26, 28, 48]).
In this paper, we consider the modified q-Daehee polynomials of the second kind and investigate their properties. Furthermore, we consider the modified degenerate q-Daehee polynomials of the second kind and investigate their properties.
2 The modified q-Daehee polynomials and numbers of the second kind
Let p be a fixed prime number. We assume that \(t \in \mathbb{C}_{p}\) with \(\vert t \vert _{p} < p^{-\frac{1}{p-1}}\) and \(q\in \mathbb{C}_{p}\) with \(\vert 1-q \vert _{p}< p^{-\frac{1}{p-1}} \).
The modified q-Daehee polynomials of the second kind are defined by
When \(x=0\), \(D_{n,q}^{*} =D_{n,q}^{*} (0)\) are called the nth modified q-Daehee numbers of the second kind. By using the binomial theorem in (2.1), we observe that
Note that the modified q-Daehee polynomials were defined by Lim in [31] as follows:
From (2.1) and (2.2), we obtain the following theorem.
Theorem 2.1
For \(n\geq 0\), we have
From (2.1), we derive that
By using (1.9) and (1.10) in Eq. (2.4), we have
Thus, by (2.1), (2.5), and (2.6), we obtain the following theorem.
Theorem 2.2
For \(n\geq 0\), we have
From (2.1), by replacing t by \(e^{t} -1\) and using (1.8), we get
and by using (1.10) and (2.3), we have
From (2.8) and (2.9), we obtain the following theorem.
Theorem 2.3
For \(n\geq 0\), we have
3 The modified degenerate q-Daehee polynomials of the second kind
Let p be a fixed prime number. We assume that \(t \in \mathbb{C}_{p}\) with \(\vert t \vert _{p} < p^{-\frac{1}{p-1}}\).
The modified degenerate q-Daehee polynomials of the second kind are defined by
When \(x=0\), \(D_{n, \lambda , q}^{*} =D_{n, \lambda ,q}^{*} (0)\) are called the modified degenerate q-Daehee numbers of the second kind.
We note that the reason for calling \(D_{n, \lambda , q}^{*}\) the modified degenerate q-Daehee polynomials of the second kind is to distinguish it from the modified q-Daehee numbers and polynomials in [31]. From (3.1), we observe that
From (3.1) and (3.2), we obtain the following theorem.
Theorem 3.1
For \(n\geq 0\), we have
From (3.1), by replacing t by \(\frac{1}{\lambda } (e^{\lambda t} -1)\), we derive
From (3.4) and (2.1), we obtain the following theorem.
Theorem 3.2
For \(n\geq 0\), we have
From (3.1), we observe that
From (3.7), we get
From (3.7) and (3.1), we obtain the following theorem.
Theorem 3.3
For \(n\geq 0\), we have
4 Conclusion
Many authors studied the q-Daehee polynomials (1.5), the degenerate λ-q-Daehee polynomials of the second kind in [12, 33, 46]. In this paper, we defined the modified q-Daehee polynomials of the second kind (2.1), which are different from the q-Daehee polynomials (1.5), and the modified degenerate q-Daehee polynomials of the second kind (3.1), which are different from the modified q-Daehee numbers and polynomials in [31]. We obtained the interesting results of Theorems 2.1, 2.2, and 2.3, which are some identity properties related with the modified degenerate q-Daehee polynomials of the second kind (3.1) and also we obtained the results of Theorems 3.1, 3.2, and 3.3, which are some identities related with the modified q-Daehee polynomials of the second kind.
References
Araci, S., Acikgoz, M.: A note on the Frobenius–Euler numbers and polynomials associated with Bernstein polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 22(3), 399–406 (2012)
Bayad, A., Chikhi, J.: Apostol–Euler polynomials and asymptotics for negative binomial reciprocals. Adv. Stud. Contemp. Math. (Kyungshang) 24(1), 33–37 (2014)
Carlitz, L.: q-Bernoulli and Eulerian numbers. Trans. Am. Math. Soc. 76, 332–350 (1954)
Carlitz, L.: q-Bernoulli numbers and polynomials. Duke Math. J. 25, 987–1000 (1958)
Carlitz, L.: Expansions of q-Bernoulli numbers. Duke Math. J. 25, 355–364 (1958)
Dolgy, D.V., Jang, G.-W., Kwon, H.-I., Kim, T.: A note on Carlitz’s type q-Changhee numbers and polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 27(4), 451–459 (2017)
Dolgy, D.V., Kim, T.: Some explicit formulas of degenerate Stirling numbers associated with the degenerate special numbers and polynomials. Proc. Jangjeon Math. Soc. 21(2), 309–317 (2018)
El-Desouky, B.S., Mustafa, A.: New results on higher-order Daehee and Bernoulli numbers and polynomials. Adv. Differ. Equ. 2016, 32 (2016)
Jang, G.-W., Kim, T.: Revisit of identities for Daehee numbers arising from nonlinear differential equations. Proc. Jangjeon Math. Soc. 20(2), 163–177 (2017)
Jang, G.W., Kim, D.S., Kim, T.: Degenerate Changhee numbers and polynomials of the second kind. Adv. Stud. Contemp. Math. (Kyungshang) 27(4), 609–624 (2017)
Khan, W.A., Nisar, K.S., Duran, U., Acikgoz, M., Araci, S.: Multifarious implicit summation formulae of Hermite-based poly-Daehee polynomials. Proc. Jangjeon Math. Soc. 21(3), 305–310 (2018)
Kim, B.M., Yun, S.J., Park, J.-W.: On a degenerate λ-q-Daehee polynomials. J. Nonlinear Sci. Appl. 9, 4607–4616 (2016)
Kim, D.S., Kim, T.: A note on degenerate Eulerian numbers and polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 27(4), 431–440 (2017)
Kim, D.S., Kim, T.: A new approach to Catalan numbers using differential equations. Russ. J. Math. Phys. 24(4), 465–475 (2018)
Kim, D.S., Kim, T.: Some p-adic integrals on \(\mathbb{Z}_{p}\) associated with trigonometric functions. Russ. J. Math. Phys. 25(3), 300–308 (2018)
Kim, D.S., Kim, T., Kwon, H.-I., Jang, G.-W.: Degenerate Daehee polynomials of the second kind. Proc. Jangjeon Math. Soc. 21(1), 83–97 (2018)
Kim, T.: On explicit formulas of p-adic \(q-L\)-functions. Kyushu J. Math. 48(1), 73–86 (1994)
Kim, T.: On p-adic q-Bernoulli numbers. J. Korean Math. Soc. 37(1), 21–30 (2000)
Kim, T.: q-Volkenborn integration. Russ. J. Math. Phys. 9(3), 288–299 (2002)
Kim, T.: An invariant p-adic q-integral on \(\mathbb{Z}_{p} \). Appl. Math. Lett. 21(2), 105–108 (2008)
Kim, T.: On degenerate q -Bernoulli polynomials. Bull. Korean Math. Soc. 53(4), 1149–1156 (2016)
Kim, T.: λ-Analogue of Stirling numbers of the first kind. Adv. Stud. Contemp. Math. (Kyungshang) 27(3), 423–429 (2017)
Kim, T.: A note on degenerate Stirling polynomials of the second kind. Proc. Jangjeon Math. Soc. 20(3), 319–331 (2017)
Kim, T.: Degenerate Cauchy numbers and polynomials of the second kind. Adv. Stud. Contemp. Math. (Kyungshang) 27(4), 441–449 (2018)
Kim, T., Jang, G.-W.: Higher-order degenerate q-Bernoulli polynomials. Proc. Jangjeon Math. Soc. 20(1), 51–60 (2017)
Kim, T., Jang, G.W.: A note on degenerate gamma function and degenerate Stirling number of the second kind. Adv. Stud. Contemp. Math. (Kyungshang) 28(2), 207–214 (2018)
Kim, T., Kim, D.S.: Degenerate Laplace transform and degenerate gamma function. Russ. J. Math. Phys. 24(2), 241–248 (2017)
Kim, T., Kim, D.S.: Identities for degenerate Bernoulli polynomials and Korobov polynomials. Sci. China Math. (2018). http://engine.scichina.com/publisher/scp/journal/SCM/doi/10.1007/s11425-018-9338-5?slug=abstract. https://doi.org/10.1007/s11425-018-9338-5
Kim, T., Simsek, Y.: Analytic continuation of the multiple Daehee \(q-l\)-functions associated with Daehee numbers. Russ. J. Math. Phys. 15(1), 58–65 (2008)
Kim, T., Yao, Y., Kim, D.S., Jang, G.-W.: Degenerate r-Stirling numbers and r-Bell polynomials. Russ. J. Math. Phys. 25(1), 44–58 (2018)
Lim, D.: Modified q-Daehee numbers and polynomials. J. Comput. Anal. Appl. 21(2), 324–330 (2016)
Liu, C., Wuyungaowa: Application of probabilistic method on Daehee sequences. Eur. J. Pure Appl. Math. 11(1), 69–78 (2018)
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)
Ozden, H., Cangul, I.N., Simsek, Y.: Remarks on q-Bernoulli numbers associated with Daehee numbers. Adv. Stud. Contemp. Math. (Kyungshang) 18(1), 41–48 (2009)
Park, J.-W.: On the q-analogue of Daehee numbers and polynomials. Proc. Jangjeon Math. Soc. 19(3), 537–544 (2016)
Park, J.-W., Kim, B.M., Kwon, J.: On a modified degenerate Daehee polynomials and numbers. J. Nonlinear Sci. Appl. 10, 1108–1115 (2017)
Pyo, S.-S.: Degenerate Cauchy numbers and polynomials of the fourth kind. Adv. Stud. Contemp. Math. (Kyungshang) 28(1), 127–138 (2018)
Rim, S.-H., Kim, T., Pyo, S.-S.: Identities between harmonic, hyperharmonic and Daehee numbers. J. Inequal. Appl. 2018, 168 (2018)
Schikhof, W.H.: Ultrametric Calculus: An Introduction to a p-Adic Analysis. Cambridge Studies in Advanced Mathematics, vol. 4, p. 167, Definition 55.1. Cambridge University Press, Cambridge (1985)
Shiratani, K., Yokoyama, S.: An application of p-adic convolutions. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math. 36(1), 73–83 (1982)
Simsek, Y.: Analysis of the p-adic q-Volkenborn integrals; an approach to generalized Apostrol-type special numbers and polynomials and their applications. Cogent Math. 3, 1269393 (2016)
Simsek, Y.: Apostol type Daehee numbers and polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 26(3), 555–566 (2016)
Simsek, Y.: Identities on the Changhee numbers and Apostol-type Daehee polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 27(2), 199–212 (2017)
Simsek, Y.: Identities and relations related to combinatorial numbers and polynomials. Proc. Jangjeon Math. Soc. 20(1), 127–135 (2017)
Simsek, Y.: Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and p-adic q-integrals. Turk. J. Math. 42, 557–577 (2018)
Simsek, Y., Rim, S.-H., Jang, L.-C., Kang, D.-J., Seo, J.-J.: A note on q-Daehee sums. In: Proceedings of the 16th. International Conference of the Jangjeon Mathematical Society, vol. 36, pp. 159–166. Jangjeon Math. Soc., Hapcheon (2005)
Simsek, Y., Yardimci, A.: Applications on the Apostol–Daehee numbers and polynomials associated with special numbers, polynomials, and p-adic integrals. Adv. Differ. Equ. 2016, 308 (2016)
Washington, L.C.: Introduction to Cyclotomic Fields, 2nd edn. Graduate Texts in Mathematics, vol. 83, xiv+487 pp. Springer, New York (1997). ISBN 0-387947620
Funding
This paper was supported by Wonkwang University in 2017.
Author information
Authors and Affiliations
Contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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
Lee, J.G., Kim, W.J., Jang, LC. et al. A note on modified degenerate q-Daehee polynomials and numbers. J Inequal Appl 2019, 24 (2019). https://doi.org/10.1186/s13660-019-1966-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-019-1966-1