Research Article
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Year 2021, , 1 - 8, 01.03.2021
https://doi.org/10.36753/mathenot.839111

Abstract

References

  • [1] Bayad, A., Hamahata, Y.: Polylogarithms and poly-Bernoulli polynomials. Kyushu J. Math. 65, 15-24 (2012).
  • [2] Carlitz, L.: Degenerate Stirling, Bernoulli and Eulerian numbers. Utilitas Math. 15, 51-88 (1979).
  • [3] Chung, S.-K., Jang, G.-W., Kim, D.S., Kwon, J.: Some identities of the type 2 degenerate Bernoulli and Euler numbers. Adv. Stud. Contemp. Math. (Kyungshang). 29 (4), 613-632 (2019).
  • [4] Duran, U., Acikgoz, M., Araci, S.: Hermite based poly-Bernoulli polynomials with a q parameter, Adv. Stud. Contemp. Math. (Kyungshang). 28 (2), 285-296 (2018).
  • [5] Kaneko, M.: poly-Bernoulli numbers. J. Theor. Nombres Bordx. 9, 221-228 (1997).
  • [6] Kumar Sharma, S., Khan,W.A., Araci, S., Ahmed, S.S.: New type of degenerate Daehee polynomials of the second kind. Adv. Differ. Equ. 428 (2020).
  • [7] Sharma, S.K., Khan, W.A., Araci, S., Ahmed, S.S.: New construction of type 2 degenerate central Fubini polynomials with their certain properties. Adv. Differ. Equ. 587 (2020).
  • [8] Kilar, N., Simsek, Y.: Relations on Bernoulli and Euler polynomials related to trigonometric functions. Adv. Stud. Contemp. Math. (Kyungshang). 29 (2), 191-198 (2019).
  • [9] Kim, T., Kim, D.S., Kwon, J., Lee, H.: Degenerate polyexponential functions and type 2 degenerate poly-Bernoulli numbers and polynomials. Adv. Differ. Equ. 168 (2020).
  • [10] Kim, T., Jang, Y.S; Seo, J.J: A note on Poly-Genocchi numbers and polynomials. Appl. Math. Sci. 8 (96), 4775-4781 (2014).
  • [11] Kim, T., Kim, D.S.: A note on type 2 Changhee and Daehee polynomials. RACSAM. 113, 2783-2791 (2019).
  • [12] Kim, D.S., Kim, T.: A note on degenerate poly-Bernoulli numbers and polynomials. Adv. Differ. Equ. 258 (2015).
  • [13] Kim, T., Kim, D.S., Kim, H.Y., Jang, L.-C.: Degenerate poly-Bernoulli numbers and polynomials. Informatica. 31 (3), 2-8 (2020).
  • [14] Kim, T., Jang, L.-C., Kim, D. S., Kim, H. Y.: Some identities on type 2 degenerate Bernoulli polynomials of the second kind. Symmetry. 12 (4), 510 (2020).
  • [15] Kim, D.S., Kim, T., Ryoo, C.S.: Generalized type 2 degenerate Euler numbers. Adv. Stud.Contemp. Math. (Kyungshang). 30 (2), 165-169, (2020).
  • [16] Kwon, J., Kim,W.J., Rim, S.-H.: On the some identities of the type 2 Daehee and Changhee polynomials arising from p-adic integrals on Zp. Proc. Jangjeon Math. Soc. 22 (3), 487-497 (2019).
  • [17] Kwon, J., Jang, L.-C.: A note on type 2 poly-Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. (Kyungshang). 30 (2), 253-262 (2020).
  • [18] Lee, D.S., Kim, H.K., Jang, L.-C.: Type 2 degenerate poly-Euler Polynomials. Symmetry. 12, 1011 (2020).
  • [19] Jang, L.-C., Kim, D.S., Kim, T., Lee, H. p-adic integral on Zp associated with degenerate Bernoulli polynomials of the second kind. Adv. Differ. Equ. 2020, 278 (2020).
  • [20] Jang, G.-W., Kim, T.: A note on type 2 degenerate Euler and Bernoulli polynomials. Adv. Stud. Contemp. Math. (Kyungshang). 29 (1), 147-159 (2019).
  • [21] Raza, N., Zainab, U., Araci, S., Esi, A.: Identities involving 3-variable Hermite polynomials arising from umbral method. Adv. Differ. Equ. 2020 (640), (2020).

Degenerate Poly-Type 2-Bernoulli Polynomials

Year 2021, , 1 - 8, 01.03.2021
https://doi.org/10.36753/mathenot.839111

Abstract

Recently, Kim-Kim [10] have studied type 2-Changhee and Daehee polynomials. They have also introduced the type 2-Bernoulli polynomials in order to express the central factorial numbers of the second kind by making use of type 2-Bernoulli numbers of negative integral orders. Inspired by their work, we consider a new class of generating functions of type 2-Bernoulli polynomials. We give some identities for these polynomials including type 2-Euler polynomials and Stirling numbers of the second kind.

References

  • [1] Bayad, A., Hamahata, Y.: Polylogarithms and poly-Bernoulli polynomials. Kyushu J. Math. 65, 15-24 (2012).
  • [2] Carlitz, L.: Degenerate Stirling, Bernoulli and Eulerian numbers. Utilitas Math. 15, 51-88 (1979).
  • [3] Chung, S.-K., Jang, G.-W., Kim, D.S., Kwon, J.: Some identities of the type 2 degenerate Bernoulli and Euler numbers. Adv. Stud. Contemp. Math. (Kyungshang). 29 (4), 613-632 (2019).
  • [4] Duran, U., Acikgoz, M., Araci, S.: Hermite based poly-Bernoulli polynomials with a q parameter, Adv. Stud. Contemp. Math. (Kyungshang). 28 (2), 285-296 (2018).
  • [5] Kaneko, M.: poly-Bernoulli numbers. J. Theor. Nombres Bordx. 9, 221-228 (1997).
  • [6] Kumar Sharma, S., Khan,W.A., Araci, S., Ahmed, S.S.: New type of degenerate Daehee polynomials of the second kind. Adv. Differ. Equ. 428 (2020).
  • [7] Sharma, S.K., Khan, W.A., Araci, S., Ahmed, S.S.: New construction of type 2 degenerate central Fubini polynomials with their certain properties. Adv. Differ. Equ. 587 (2020).
  • [8] Kilar, N., Simsek, Y.: Relations on Bernoulli and Euler polynomials related to trigonometric functions. Adv. Stud. Contemp. Math. (Kyungshang). 29 (2), 191-198 (2019).
  • [9] Kim, T., Kim, D.S., Kwon, J., Lee, H.: Degenerate polyexponential functions and type 2 degenerate poly-Bernoulli numbers and polynomials. Adv. Differ. Equ. 168 (2020).
  • [10] Kim, T., Jang, Y.S; Seo, J.J: A note on Poly-Genocchi numbers and polynomials. Appl. Math. Sci. 8 (96), 4775-4781 (2014).
  • [11] Kim, T., Kim, D.S.: A note on type 2 Changhee and Daehee polynomials. RACSAM. 113, 2783-2791 (2019).
  • [12] Kim, D.S., Kim, T.: A note on degenerate poly-Bernoulli numbers and polynomials. Adv. Differ. Equ. 258 (2015).
  • [13] Kim, T., Kim, D.S., Kim, H.Y., Jang, L.-C.: Degenerate poly-Bernoulli numbers and polynomials. Informatica. 31 (3), 2-8 (2020).
  • [14] Kim, T., Jang, L.-C., Kim, D. S., Kim, H. Y.: Some identities on type 2 degenerate Bernoulli polynomials of the second kind. Symmetry. 12 (4), 510 (2020).
  • [15] Kim, D.S., Kim, T., Ryoo, C.S.: Generalized type 2 degenerate Euler numbers. Adv. Stud.Contemp. Math. (Kyungshang). 30 (2), 165-169, (2020).
  • [16] Kwon, J., Kim,W.J., Rim, S.-H.: On the some identities of the type 2 Daehee and Changhee polynomials arising from p-adic integrals on Zp. Proc. Jangjeon Math. Soc. 22 (3), 487-497 (2019).
  • [17] Kwon, J., Jang, L.-C.: A note on type 2 poly-Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. (Kyungshang). 30 (2), 253-262 (2020).
  • [18] Lee, D.S., Kim, H.K., Jang, L.-C.: Type 2 degenerate poly-Euler Polynomials. Symmetry. 12, 1011 (2020).
  • [19] Jang, L.-C., Kim, D.S., Kim, T., Lee, H. p-adic integral on Zp associated with degenerate Bernoulli polynomials of the second kind. Adv. Differ. Equ. 2020, 278 (2020).
  • [20] Jang, G.-W., Kim, T.: A note on type 2 degenerate Euler and Bernoulli polynomials. Adv. Stud. Contemp. Math. (Kyungshang). 29 (1), 147-159 (2019).
  • [21] Raza, N., Zainab, U., Araci, S., Esi, A.: Identities involving 3-variable Hermite polynomials arising from umbral method. Adv. Differ. Equ. 2020 (640), (2020).
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Serkan Araci 0000-0002-3950-6864

Publication Date March 1, 2021
Submission Date December 11, 2020
Acceptance Date February 13, 2021
Published in Issue Year 2021

Cite

APA Araci, S. (2021). Degenerate Poly-Type 2-Bernoulli Polynomials. Mathematical Sciences and Applications E-Notes, 9(1), 1-8. https://doi.org/10.36753/mathenot.839111
AMA Araci S. Degenerate Poly-Type 2-Bernoulli Polynomials. Math. Sci. Appl. E-Notes. March 2021;9(1):1-8. doi:10.36753/mathenot.839111
Chicago Araci, Serkan. “Degenerate Poly-Type 2-Bernoulli Polynomials”. Mathematical Sciences and Applications E-Notes 9, no. 1 (March 2021): 1-8. https://doi.org/10.36753/mathenot.839111.
EndNote Araci S (March 1, 2021) Degenerate Poly-Type 2-Bernoulli Polynomials. Mathematical Sciences and Applications E-Notes 9 1 1–8.
IEEE S. Araci, “Degenerate Poly-Type 2-Bernoulli Polynomials”, Math. Sci. Appl. E-Notes, vol. 9, no. 1, pp. 1–8, 2021, doi: 10.36753/mathenot.839111.
ISNAD Araci, Serkan. “Degenerate Poly-Type 2-Bernoulli Polynomials”. Mathematical Sciences and Applications E-Notes 9/1 (March 2021), 1-8. https://doi.org/10.36753/mathenot.839111.
JAMA Araci S. Degenerate Poly-Type 2-Bernoulli Polynomials. Math. Sci. Appl. E-Notes. 2021;9:1–8.
MLA Araci, Serkan. “Degenerate Poly-Type 2-Bernoulli Polynomials”. Mathematical Sciences and Applications E-Notes, vol. 9, no. 1, 2021, pp. 1-8, doi:10.36753/mathenot.839111.
Vancouver Araci S. Degenerate Poly-Type 2-Bernoulli Polynomials. Math. Sci. Appl. E-Notes. 2021;9(1):1-8.

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