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Some identities on the Higher-order Daehee and Changhee Numbers

Received: 1 June 2015     Accepted: 16 June 2015     Published: 5 August 2015
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Abstract

In this note, we shall give an explicit formula for the coefficients of the expansion of given generating function, when that function has an appropriate form, the coefficients can be represented by the higher-order Daehee and Changhee polynomials and numbers of the first kind. By the classical method of comparing the coefficients of the generating function, we show some interesting identities related to the Higher-order Daehee and Changhee numbers.

Published in Pure and Applied Mathematics Journal (Volume 4, Issue 5-1)

This article belongs to the Special Issue Mathematical Aspects of Engineering Disciplines

DOI 10.11648/j.pamj.s.2015040501.17
Page(s) 33-37
Creative Commons

This is an Open Access article, 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 or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Higher-order Daehee Numbers, Higher-order Changhee Numbers, Bernoulli Number, Euler Number

References
[1] D. S. Kim, T. Kim, S.-H. Lee, and J.-J. Seo, Higher-order Daehee numbers and polynomials, International Journal of Mathematical Analysis 8 (2014), no. 5-6,273–283.
[2] D. S. Kim, T. Kim, and J.-J. Seo, Higher-order Daehee polynomials of the first kind with umbral calculus, Adv. Stud. Contemp. Math. (Kyungshang) 24(2014), no. 1, 5–18. MR 3157404
[3] D. S. Kim, T. Kim, J.-J. Seo, and S.-H. Lee, Higher-order Changhee numbers and polynomials, Adv. Studies Theor. Phys. 8 (2014), no. 8, 365–373.
[4] D. S. Kim, T. Kim, Higher-order Cauchy of the first kind and poly-Cauchy of the first kind mixed type polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 23 (2013), no. 4, 621–636.
[5] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, (Translated from the French by J. W. ienhuys), Reidel, Dordrecht, 1974.
[6] G.-D. Liu and H. M. Srivastava, Explicit formulas for the Nörlund polynomials of the first and second kind, Comput. Math. Appl. 51 (2006), no. 9–10,1377–1384.
[7] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), no. 3, 288–299. MR 1965383 (2004f:11138).
[8] T. Kim, Symmetry p-adic invariant integral on Zp for Bernoulli and Euler polynomials, J. Difference Equ. Appl. 14 (2008), no. 12, 1267–1277.MR 2462529 (2009i:11023)
[9] Z. Zhang and H. Yang, Some closed formulas for generalized Bernoulli-Euler numbers and polynomials, Proc. Jangjeon Math. Soc. 11 (2008), no. 2, 191–198. MR 2482602 (2010a:11036).
[10] D. Ding and J. Yang, Some identities related to the Apostol -Euler and Apostol Bernoulli polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010).no. 1, 7–21. MR 2597988 (2011k:05030).
[11] K.-W. Hwang, D. V. Dolgy, D. S. Kim, T. Kim, and S. H. Lee, Some theorems on Bernoulli and Euler numbers, Ars Combin. 109 (2013), 285–297.MR 3087218.
[12] H. M. Srivastava and J.-S. Choi, Series Associated with the Zeta and Related Functions, Kluwer Acad. Publ., Dordrecht, oston and London,2001.
[13] G. -D. Liu,Generating functions and generalized Euler numbers, Proc. Japan Acad., 84, Ser. A (2008),29-34.
[14] D.S. Kim, T. Kim, Identities of some special mixed-type polynomials, arXiv:1406.2124v1 [math.NT] 9 Jun 2014.
Cite This Article
  • APA Style

    Nian Liang Wang, Hailong Li. (2015). Some identities on the Higher-order Daehee and Changhee Numbers. Pure and Applied Mathematics Journal, 4(5-1), 33-37. https://doi.org/10.11648/j.pamj.s.2015040501.17

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    ACS Style

    Nian Liang Wang; Hailong Li. Some identities on the Higher-order Daehee and Changhee Numbers. Pure Appl. Math. J. 2015, 4(5-1), 33-37. doi: 10.11648/j.pamj.s.2015040501.17

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    AMA Style

    Nian Liang Wang, Hailong Li. Some identities on the Higher-order Daehee and Changhee Numbers. Pure Appl Math J. 2015;4(5-1):33-37. doi: 10.11648/j.pamj.s.2015040501.17

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  • @article{10.11648/j.pamj.s.2015040501.17,
      author = {Nian Liang Wang and Hailong Li},
      title = {Some identities on the Higher-order Daehee and Changhee Numbers},
      journal = {Pure and Applied Mathematics Journal},
      volume = {4},
      number = {5-1},
      pages = {33-37},
      doi = {10.11648/j.pamj.s.2015040501.17},
      url = {https://doi.org/10.11648/j.pamj.s.2015040501.17},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2015040501.17},
      abstract = {In this note, we shall give an explicit formula for the coefficients of the expansion of given generating function, when that function has an appropriate form, the coefficients can be represented by the higher-order Daehee and Changhee polynomials and numbers of the first kind. By the classical method of comparing the coefficients of the generating function, we show some interesting identities related to the Higher-order Daehee and Changhee numbers.},
     year = {2015}
    }
    

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    AB  - In this note, we shall give an explicit formula for the coefficients of the expansion of given generating function, when that function has an appropriate form, the coefficients can be represented by the higher-order Daehee and Changhee polynomials and numbers of the first kind. By the classical method of comparing the coefficients of the generating function, we show some interesting identities related to the Higher-order Daehee and Changhee numbers.
    VL  - 4
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Author Information
  • Department of Applied Mathematics, Shangluo University, Shangluo, Peoples Republic of China

  • Department of Mathematics and Information Science, Weinan Normal University, Weinan, Peoples Republic of China

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