Degenerate Poly-Type 2-Bernoulli Polynomials
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.
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- [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).