Explicit expression for symmetric identities of w-Catalan–Daehee polynomials

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 ℤp.

Keywords

  • Catalan numbers
  • Daehee numbers
  • w-Catalan–Daehee numbers

2010 Mathematics Subject Classification

  • 11B83
  • 11S80

References

  1. Choi, S. (2018) Linear symmetry of the modified q-Euler polynomials. Adv. Stud. Contemp. Math. (Kyungshang), 28 (2), 201–206.
  2. 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.
  3. 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.
  4. Duran, U., Acikgoz, M., & Araci, S. (2015) Symmetric identities involving weighted q-Genocchi polynomials under S4. Proc. Jangjeon Math. Soc., 18 (4), 455–465.
  5. 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.
  6. He, Y. (2013) Symmetric identities for Carlitz’s q-Bernoulli numbers and polynomials. Adv. Difference Equ., 2013:246, 10 pages.
  7. 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.
  8. 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.
  9. 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.
  10. Kim, D. S., & Kim, T. (2017) Triple symmetric identities for w-Catalan polynomials, J. Korean Math. Soc., 54 (4), 1243–1264.
  11. 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.
  12. Kim, T. (2016) A note in Catalan numbers associated with p-adic integral in p, Proc. Jangjeon Math. Soc., 19 (3), 493–501.
  13. 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.
  14. 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.
  15. Koshy, T. (2009) Catalan Numbers with Applications. Oxford University Press, Oxford.
  16. 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.
  17. Lim, D., Kwon, J. (2016) A note on poly-Daehee numbers and polynomials. Proc. Jangjeon Math. Soc., 19 (2), 219—224.
  18. Liu, C., & Wuyungaowa, W. (2018) Application of probabilistic method on Daehee sequences, Eur. J. Pure Appl. Math., 11 (1), 69–78.
  19. 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.
  20. Park, J.-W. (2016) On the λ-Daehee polynomials with q-parameter. J. Comput. Anal. Appl., 20 (1), 11–20.
  21. 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.
  22. Pyo, S.-S., Kim, T., & Rim, S.-H. Degenerate Daehee numbers of the third kind (submitted).
  23. Simsek, Y. (2017) Identities on the Changhee numbers and Apostol-type Daehee polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 27 (2), 199–212.
  24. 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.
  25. 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.

Comments are closed.