Taekyun Kim, Seog-Hoon Rim, Dmitry V. Dolgy and Sung-Soo Pyo

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 24, 2018, Number 4, Pages 99—111

DOI: 10.7546/nntdm.2018.24.4.99-111

**Download full paper: PDF, 133 Kb**

## Details

### Authors and affiliations

Taekyun Kim

*Department of Mathematics, Tianjin Polytechnic University
Tianjin 300387, China, and
Department of Mathematics, Kwangwoon University
Seoul, Republic of Korea*

Seog-Hoon Rim

*Department of Mathematics Education, Kyungpook National University
Daegu, Republic of Korea*

Dmitry V. Dolgy

*Institute of Natural Sciences, Far Eastern Federal University
Vladivostok, 690950, Russia*

Sung-Soo Pyo

*Department of Mathematics Education, Silla University
Busan, Republic of Korea*

### Abstract

Recently, Catalan–Daehee numbers are studied by several authors. In this paper, we consider the *w*-Catalan–Daehee polynomials and investigate some properties for those polynomials. In addition, we give explicit expression for the symmetric identities of the *w*-Catalan–Daehee polynomials which are derived from p-adic invariant integral on ℤ_{p}.

### Keywords

- Catalan numbers
- Daehee numbers
*w*-Catalan–Daehee numbers

### 2010 Mathematics Subject Classification

- 11B83
- 11S80

### References

- Choi, S. (2018) Linear symmetry of the modified q-Euler polynomials. Adv. Stud. Contemp. Math. (Kyungshang), 28 (2), 201–206.
- Choi, S., Kim, T., Kwon, H.-I., & Kwon, J. (2018) Quadratic symmetry of modified q-Euler polynomials, Adv. Difference Equ., Paper No. 38, 9 pages.
- Dolgy, D. V., Jang, G.-W., Kim, D. S., & Kim, T. (2017) Explicit Expressions for Catalan–Daehee numbers, Proc. Jangjeon Math. Soc., 20(1), 1–9.
- Duran, U., Acikgoz, M., & Araci, S. (2015) Symmetric identities involving weighted
*q*-Genocchi polynomials under S4. Proc. Jangjeon Math. Soc., 18 (4), 455–465. - El-Desouky, B. S., & Mustafa, A. (2016) New results on higher-order Daehee and Bernoulli numbers and polynomials, Adv. Difference Equ., 2016, Paper No. 32, 21 pages.
- He, Y. (2013) Symmetric identities for Carlitz’s q-Bernoulli numbers and polynomials. Adv. Difference Equ., 2013:246, 10 pages.
- Jang, G.-W., Kwon, J., & Lee, J. G. (2017) Some identities of degenerate Daehee numbers arising from nonlinear differential equation. Adv. Difference Equ., 2017, Paper No. 206, 10 pages.
- Jang, L.-C. (2011) A family of Barnes-type multiple twisted q -Euler numbers and polynomials related to Fermionic p-adic invariant integrals on Zp, J. Comput. Anal. Appl., 13 (2), 376–387.
- Khan, W. A., Nisar, K. S., Duran, U., Acikgoz, M., & Araci, S. (2018) Multifarious implicit summation formulae of Hermite-based poly-Daehee polynomials, Appl. Math. Inf. Sci., 12(2), 305–310.
- Kim, D. S., & Kim, T. (2017) Triple symmetric identities for w-Catalan polynomials, J. Korean Math. Soc., 54 (4), 1243–1264.
- Kim, D. S., Lee, N., Na, J., & Park, K. H. (2012) Identities of symmetry for higher-order Euler polynomials in three variables (I). Adv. Stud. Contemp. Math. (Kyungshang), 22 (1), 51–74.
- Kim, T. (2016) A note in Catalan numbers associated with p-adic integral in
**ℤ**_{p}, Proc. Jangjeon Math. Soc., 19 (3), 493–501. - Kim, T., & Kim, D. S. (2017) Differential equations associated with Catalan–Daehee numbers and their applications, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 111 (4), 1071–1081.
- Kim, T., Kim, D. S., & Seo, J.-J. (2016) Symmetric identities for an analogue of Catalan polynomials, Proc. Jangjeon Math. Soc., 19 (3), 515–521.
- Koshy, T. (2009) Catalan Numbers with Applications. Oxford University Press, Oxford.
- Kwon, J., Sohn, G., & Park, J.-W. (2018) Symmetric identities for (
*h*,*q*)-extensions of the generalized higher order modified q-Euler polynomials. J. Comput. Anal. Appl., 24 (8), 1431–1438. - Lim, D., Kwon, J. (2016) A note on poly-Daehee numbers and polynomials. Proc. Jangjeon Math. Soc., 19 (2), 219—224.
- Liu, C., & Wuyungaowa, W. (2018) Application of probabilistic method on Daehee sequences, Eur. J. Pure Appl. Math., 11 (1), 69–78.
- Moon, E.-J., Rim, S.-H., Jin, J.-H., & Lee, S.-J. (2010) On the symmetric properties of higher-order twisted
*q*-Euler numbers and polynomials, Adv. Difference Equ., 2010, Art. ID 765259, 8 pages. - Park, J.-W. (2016) On the λ-Daehee polynomials with
*q*-parameter. J. Comput. Anal. Appl., 20 (1), 11–20. - Pyo, S.-S., Kim, T., & Rim, S.-H. (2017) Identities of the degenerate Daehee numbers with the Bernoulli numbers of the second kind arising from nonlinear differential equation, J. Nonlinear Sci. Appl., 10, 6219–6228.
- Pyo, S.-S., Kim, T., & Rim, S.-H. Degenerate Daehee numbers of the third kind (submitted).
- Simsek, Y. (2017) Identities on the Changhee numbers and Apostol-type Daehee polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 27 (2), 199–212.
- Simsek, Y., & Yardimci, A. (2016) Applications on the Apostol–Daehee numbers and polynomials associated with special numbers, polynomials, and
*p*-adic integrals. Adv. Difference Equ., 2016, Paper No. 308, 14 pages. - Stanley, R. P. (2015) Catalan numbers. Cambridge University Press, New York.

## Related papers

## Cite this paper

Ediz, S., & Semiz, M. (2018). Explicit expression for symmetric identities of *w*-Catalan–Daehee polynomials. *Notes on Number Theory and Discrete Mathematics*, 24(4), 99-111, doi: 10.7546/nntdm.2018.24.4.99-111.