Applied and Computational Mathematics

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Some Identities Related with the Higher-order Deformed Degenerate Bernoulli and Euler Polynomials

Received: Jul. 25, 2017    Accepted: Nov. 10, 2017    Published: Dec. 15, 2017
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Abstract

Recently, Kim-Kim (2016-2017) studied simmetric identities of higher-order degenerate Bernoulli and Euler polynomials which were defined by Carlitz (1979). In this paper, we define the higher-order deformed degenerate Bernoulli and Euler polynomials which are modified the higher-order degenerate Bernoulli and Euler polynomials. We also investigate some interesting identities for the the higher-order deformed degenerate Bernoulli and Euler polynomials.

DOI 10.11648/j.acm.20170606.13
Published in Applied and Computational Mathematics ( Volume 6, Issue 6, December 2017 )
Page(s) 254-258
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), 2024. Published by Science Publishing Group

Keywords

Bernoulli Polynomials, Euler Polynomials, Degenerate Bernoulli Polynomials

References
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[2] L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math. (Basel), 7 (1956), 28-33.
[3] T. Kim, D. S. Kim, Degenerate Laplace transform and degenerate gamma function, Russ. J. Math. Phys., 24(2) (2017), 241-248.
[4] D. S. Kim, T. Kim, On degenerate Bell numbers and polynomials, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 111(2) (2017), 435-446.
[5] D. S. Kim, T. Kim, Some identities of degenerate Daehee numbers arising from certain differential equations, J. Nonlinear Sci. Appl., 10(2) (2017), 744-751.
[6] T. Kim, V. D. Dolgy, H. I. Kwon, Expansions of degenerate q -Euler numbers and polynomials, Proc. Jangjeon Math. Soc., 19(4) (2016), 625-630.
[7] V. D. Dolgy, T. Kim, H. I. Kwon, J. J. Seo, Some identities for degenerate Euler numbers and polynomials arising from degenerate Bell polynomials, Proc. Jangjeon Math. Soc., 19(3) (2016), 457-464.
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[9] T. Kim, D. V. Dolgy, L. C. Jang, H. I. Kwon, Some identities of degenerate q –Euler polynomials under the symmetry group of degree n, J. Nonlinear Sci. Appl., 9(6) (2016), 4707-4712.
[10] T. Kim, D. S. Kim, H. I. Kwon, J. J. Seo, Some identities for degenerate Frobenius-Euler numbers arising from nonlinear differential equations, Ital. J. Pure Appl. Math., 36 (2016), 843-850.
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[14] H. I. Kwon, T. Kim, J. J. Seo, Some new identities of symmetry for modified degenerate Euler polynomials, Proc. Jangjeon Math. Soc., 19(2) (2016), 237-242.
[15] T. Kim, D. S. Kim, H. I. Kwon, Some identities relating to degenerate Bernoulli polynomials, Filomat, 30(4) (2016), 905-912.
[16] T. Kim, D. S. Kim, H. I. Kwon, J. J. Seo, Differential equations arising from the generating function of general modified degenerate Euler numbers, Adv. Difference Equ., 2016:129 (2016), 7 pp.
[17] D. S. Kim, T. Kim, D. V. Dolgy, On Carlitz’s degenerate Euler numbers and polynomials, J. Comput. Anal. Appl., 21(4) (2016), 738-742.
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[19] T. Kim, D. S. Kim, Identities involving degenerate Euler numbers and polynomials arising from non-linear differential equations, J. Nonlinear Sci. Appl., 9(5) (2016), 2086-2098.
[20] D. V. Dolgy, D. S. Kim, T. Kim, T. Mansour, Degenerate poly-Bernoulli polynomials of the second kind, J. Comput. Anal. Appl., 21(5) (2016), 954-966.
[21] D. S. Kim, T. Kim, T. Mansour, J. J. Seo, Degenerate Mittag-Leffler polynomials, Appl. Math. Comput., 274 (2016), 258-266.
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    Lee Chae Jang. (2017). Some Identities Related with the Higher-order Deformed Degenerate Bernoulli and Euler Polynomials. Applied and Computational Mathematics, 6(6), 254-258. https://doi.org/10.11648/j.acm.20170606.13

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    ACS Style

    Lee Chae Jang. Some Identities Related with the Higher-order Deformed Degenerate Bernoulli and Euler Polynomials. Appl. Comput. Math. 2017, 6(6), 254-258. doi: 10.11648/j.acm.20170606.13

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    AMA Style

    Lee Chae Jang. Some Identities Related with the Higher-order Deformed Degenerate Bernoulli and Euler Polynomials. Appl Comput Math. 2017;6(6):254-258. doi: 10.11648/j.acm.20170606.13

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  • @article{10.11648/j.acm.20170606.13,
      author = {Lee Chae Jang},
      title = {Some Identities Related with the Higher-order Deformed Degenerate Bernoulli and Euler Polynomials},
      journal = {Applied and Computational Mathematics},
      volume = {6},
      number = {6},
      pages = {254-258},
      doi = {10.11648/j.acm.20170606.13},
      url = {https://doi.org/10.11648/j.acm.20170606.13},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.acm.20170606.13},
      abstract = {Recently, Kim-Kim (2016-2017) studied simmetric identities of higher-order degenerate Bernoulli and Euler polynomials which were defined by Carlitz (1979). In this paper, we define the higher-order deformed degenerate Bernoulli and Euler polynomials which are modified the higher-order degenerate Bernoulli and Euler polynomials. We also investigate some interesting identities for the the higher-order deformed degenerate Bernoulli and Euler polynomials.},
     year = {2017}
    }
    

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    Y1  - 2017/12/15
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    N1  - https://doi.org/10.11648/j.acm.20170606.13
    DO  - 10.11648/j.acm.20170606.13
    T2  - Applied and Computational Mathematics
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    UR  - https://doi.org/10.11648/j.acm.20170606.13
    AB  - Recently, Kim-Kim (2016-2017) studied simmetric identities of higher-order degenerate Bernoulli and Euler polynomials which were defined by Carlitz (1979). In this paper, we define the higher-order deformed degenerate Bernoulli and Euler polynomials which are modified the higher-order degenerate Bernoulli and Euler polynomials. We also investigate some interesting identities for the the higher-order deformed degenerate Bernoulli and Euler polynomials.
    VL  - 6
    IS  - 6
    ER  - 

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Author Information
  • Graduate School of Education, Konkuk University, Seoul, Republic of Korea

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