Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 9520269, 5 pages
https://doi.org/10.1155/2018/9520269

Research Article

On Carlitz’s Type Modified Degenerate -Changhee Polynomials and Numbers

Byung Moon Kim,¹ Lee-Chae Jang,² Wonjoo Kim,³ and Hyuck-In Kwon⁴

¹Department of Mechanical System Engineering, Dongguk University, Gyeongju 780-714, Republic of Korea
²Graduate School of Education, Konkuk University, Seoul 143-701, Republic of Korea
³Department of Applied Mathematics, Kyung Hee University, Seoul 02447, Republic of Korea
⁴Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Correspondence should be addressed to Lee-Chae Jang; lcjang@konkuk.ac.kr

Received 17 January 2018; Accepted 22 March 2018; Published 6 June 2018

Academic Editor: J. R. Torregrosa

Copyright © 2018 Byung Moon Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Recently, Dolgy-Jang-Kwon-Kim introduced Carlitz’s type -Changhee polynomials. In this paper, we define Carlitz’s type modified degenerate -Changhee polynomials and investigate some interesting identities of these polynomials.

1. Introduction

Let be a prime number with . Throughout this paper, , , and denote the ring of -adic integers, the field of -adic rational numbers, and the completion of the algebraic closure of , respectively. The -adic norm is normalized as . Let be an indeterminate in such that . The -analogue of number is defined as . As is well known, the Euler polynomials are defined by the generating function to be

(see [1–5]).

When , are called the Euler numbers.

Recall that Carlitz considered the -analogue of Euler numbers which are given by the recurrence relation as follows:

with the usual convention about replacing by , and that he also considered -Euler polynomials which are defined by

(see [1–5]).

Let be the space of continuous -valued functions on . For , the fermionic -adic -integrals on are defined by Kim to be

where (see [1, 3, 4, 6–14]). From (2), he derived the following formula for Carlitz’s -Euler numbers.

The Changhee polynomials are defined by the generating function to be

(see [15–23]).

In [16, 24], the authors (2017) obtained that

(see [15, 16, 18–23]), where is the Stirling numbers of the first kind and is the Stirling numbers of the second kind as follows:

(see [2, 10, 25–27]).

The degenerate Euler polynomials are defined by the generating function to be

(see [3, 4]).

The degenerate -Euler polynomials are defined by the generating function to be

(see [3, 4]).

In [2, 26–28], Kim et al. (2017) defined the degenerate Stirling numbers of the second kind as follows:

where and . In [15], by using the fermionic -adic -integral on , the authors defined Carlitz’s type -Changhee polynomials as follows:

(see [1, 2, 14, 15, 24, 29–31]).

In this paper, we define Carlitz’s type modified degenerate -Changhee polynomials and investigate some interesting identities of these polynomials.

2. Carlitz’s Type Modified Degenerate -Changhee Polynomials

In this section, we assume that with and . From (4) and (6), we note that

In the viewpoint of (12) and (14), Carlitz’s type modified degenerate -Changhee polynomials are defined by

We observe that

From (15) and (16), we get

Thus, by (17), we get the following theorem.

Theorem 1. For , one has

Note that

Replacing by in (20), we observe that

From (15) and (21), we get the following theorem.

Theorem 2. For , one has

Replacing by in (15), we get

Note that

From (23) and (24), we get the following theorem.

Theorem 3. For , one has

When , are called Carlitz’s type modified degenerate -Changhee number. We also observe that

From (26), we get

By (27), we get the following theorem.

Theorem 4. For , one has

3. Results and Discussions

This study was to define the modified degenerate -Changhee polynomials in (15). Theorem 1 is an interesting identity between the modified degenerate -Changhee polynomials and the -Euler polynomials. Theorem 2 is that the modified degenerate -Changhee polynomials is represented by a sum of products of the degenerate Stirling numbers of the second kind, the Stirling numbers of the first kind, and -Changhee polynomials. Theorem 3 is an identity between the degenerate -Euler polynomials and the modified degenerate -Changhee polynomials. Theorem 4 is an identity between the modified degenerate -Changhee polynomials and the -Euler polynomials. In the future, we will study to define and to investigate the higher-order modified degenerate -Changhee polynomials (see [3, 11]) and to investigate the symmetric identities of the modified degenerate -Changhee polynomials (see [3, 11]).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Acknowledgments

The authors would like to express their gratitude for the valuable comments and suggestions of Professor Dae San Kim.

References

S. Araci and O. Ozer, “Extended q-Dedekind-type Daehee-Changhee sums associated with extended q-Euler polynomials,” Advances in Difference Equations, vol. 2015, p. 272, 2015.

L. Carlitz, “Degenerate Stirling, Bernoulli and Eulerian numbers,” Utilitas Mathematica, vol. 15, no. 2, pp. 51–88, 1979.

D. S. Kim and T. Kim, “Symmetric identities of higher-order degenerate q-Euler polynomials,” Journal of Nonlinear Sciences and Applications. JNSA, vol. 9, no. 2, pp. 443–451, 2016.

T. Kim, D. V. Dolgy, and H.-I. Kwon, “Expansions of degenerate q-Euler numbers and polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 19, no. 4, pp. 625–630, 2016.

C. S. Ryoo, “Some identities of the twisted q-Euler numbers and polynomials associated with q-Bernstein polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 14, no. 2, pp. 239–248, 2011.

D. Lim, “Fourier series of higher-order Daehee and Changhee functions and their applications,” Journal of Inequalities and Applications, vol. 2017, 150 pages, 2017.

B. M. Kim, L.-C. Jang, W. Kim, and H.-I. Kwon, “Degenerate Changhee-Genocchi numbers and polynomials,” Journal of Inequalities and Applications, vol. 2017, p. 294, 2017.

D. S. Kim, T. Kim, and D. V. Dolgy, “On q-analogs of degenerate Bernoulli polynomials,” Adv. Difference Equ, vol. 2015, p. 194, 2015.

T. Kim, “On degenerate q -Bernoulli polynomials,” Bulletin of the Korean Mathematical Society, vol. 53, no. 4, pp. 1149–1156, 2016.

T. Kim, H.-I. Kwon, and J. J. Seo, “Degenerate q-Changhee polynomials,” Journal of Nonlinear Sciences and Applications. JNSA, vol. 9, no. 5, pp. 2389–2393, 2016.

T. Kim, H.-I. Kwon, and J.-J. Seo, “Identities of symmetry for degenerate q-Bernoulli polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 18, no. 4, pp. 495–499, 2015.

J. G. Lee and J. Kwon, “The modified degenerate q-Bernoulli polynomials arising from p-adic invariant integral on Zp,” Advances in Difference Equations, vol. 2017, no. 29, p. 9, 2017.

C. Liu and W. Wuyungaowa, “Application of probabilistic method on Daehee sequences,” European Journal of Pure and Applied Mathematics, vol. 11, no. 1, pp. 69–78, 2018.

F. Qi, L.-C. Jang, and H.-I. Kwon, “Some new and explicit identities related with the Appell-type degenerate q-Changhee polynomials,” Advances in Difference Equations, p. 180, 2016.

V. D. Dolgy, G. W. Jang, H. I. Kwon, and T. Kim, “A note on Carlitzs type q-Changhee numbers and polynomials,” Adv. Stud. Contemp. Math, vol. 27, no. 4, pp. 451–459, 2017.

L.-C. Jang, C. S. Ryoo, J. J. Seo, and H. . Kwon, “Some properties of the twisted Changhee polynomials and their zeros,” Applied Mathematics and Computation, vol. 274, pp. 169–177, 2016.

J. Kwon and J.-W. Park, “On modified degenerate Changhee polynomials and numbers,” Journal of Nonlinear Sciences and Applications. JNSA, vol. 9, no. 12, pp. 6294–6301, 2016.

J. G. Lee, L.-C. Jang, J.-J. Seo, S.-K. Choi, and H. I. Kwon, “On Appell-type Changhee polynomials and numbers,” Advances in Difference Equations, vol. 2016, 2016.

J. W. Park, G. W. Jang, and J. Kwon, “The λ-analogue degenerate Changhee polynomialsand numbers,” Global J. Pure Appl.Math, vol. 13, no. 3, p. 893, 2017.

S.-H. Rim, J.-W. Park, S.-S. Pyo, and J. Kwon, “Then n-th twisted Changhee polynomials and numbers,” Bulletin of the Korean Mathematical Society, vol. 52, no. 3, pp. 741–749, 2015.

Y. Simsek and A. Yardimci, “Applications on the Apostol-Daehee numbers and polynomials associated with special numbers, polynomials, and p-adic integrals,” Advances in Difference Equations, p. 308, 2016.

Y. Simsek, “Identities on the Changhee numbers and Apostol-type Daehee polynomials<,” Adv. Stud. Contemp. Math. (Kyungshang), vol. 27, no. 2, pp. 199–212, 2017.

N. L. Wang and H. Li, “Some identities on the Higher-order Daehee and Changhee Numbers,” Pure Appl. Math. J, vol. 4, pp. 33–37, 2015.

S. K. Chung and J. K. Kwon, “Symmetry Identities of q-Changhee Polynomials,” Int. J. Appl. Engin. Research, vol. 12, no. 21, pp. 11361–11364, 2017.

D. S. Kim and T. Kim, “On degenerate Bell numbers and polynomials,” Revista de la Real Academia de Ciencias Exactas, vol. 111, no. 2, pp. 435–446, 2017.

T. Kim, “A note on degenerate Stirling polynomials of the second kind,” Proceedings of the Jangjeon Mathematical Society. Memoirs of the Jangjeon Mathematical Society, vol. 20, no. 3, pp. 319–331, 2017.

T. Kim, Y. Yao, D. S. Kim, and G.-W. Jang, “Degenerate r-Stirling Numbers and r-Bell Polynomials,” Russian Journal of Mathematical Physics, vol. 25, no. 1, pp. 44–58, 2018.

T. Kim and D. S. Kim, “Degenerate Laplace transform and degenerate gamma function,” Russian Journal of Mathematical Physics, vol. 24, no. 2, pp. 241–248, 2017.

L. Carlitz, “A degenerate Staudt-Clausen theorem,” Archiv der Mathematik, vol. 7, pp. 28–33, 1956.

O. C. Havare and H. Menken, “A note on the p-adic gamma function and q-Changhee polynomials,” Journal of Mathematics and Computer Science, vol. 18, no. 01, pp. 11–17, 2017.

J.-W. Park, “On the twisted q-Changhee polynomials of higher order,” Journal of Computational Analysis and Applications, vol. 20, no. 3, pp. 424–431, 2016.