Fully degenerate Bernoulli numbers and polynomials

• Autorzy: Kim Taekyun , Kim Dae San , Park Jin-Woo

Identyfikatory

DOI

10.1515/dema-2022-0160

• Języki publikacji: EN

Abstrakty

EN

The aim of this article is to study the fully degenerate Bernoulli polynomials and numbers, which are a degenerate version of Bernoulli polynomials and numbers and arise naturally from the Volkenborn integral of the degenerate exponential functions on Zp . We find some explicit expressions for the fully degenerate Bernoulli polynomials and numbers in terms of the degenerate Stirling numbers of the second kind, the degenerate r -Stirling numbers of the second kind, and the degenerate Stirling polynomials. We also consider the degenerate poly-Bernoulli polynomials and derive explicit representations for them in terms of the same degenerate Stirling numbers and polynomials.

Słowa kluczowe

EN

fully degenerate Bernoulli polynomials; degenerate poly-Bernoulli polynomials; degenerate Stirling numbers of the second kind; degenerate Stirling polynomials

PL

wielomiany Bernoulliego; liczby Stirlinga drugiego rodzaju; wielomiany Stirlinga

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2022
• Tom: Vol. 55, nr 1
• Strony: 604--614
• Opis fizyczny: Bibliogr. 23 poz.

Twórcy

autor

Kim Taekyun

• Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

autor

Kim Dae San

• Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

autor

Park Jin-Woo

• Department of Mathematics Education, Daegu University, Gyeongsangbuk-do, 712-714, Republic of Korea

Bibliografia

• [1] D. S. Kim and T. Kim, A note on a new type of degenerate Bernoulli numbers, Russ. J. Math. Phys. 27 (2020), no. 2, 227–235.
• [2] T. Kim and D. S. Kim, Some identities on truncated polynomials associated with degenerate Bell polynomials, Russ. J. Math. Phys. 28 (2021), no. 3, 342–355.
• [3] Y. Ma and T. Ma, A note on negative λ-binomial distribution, Adv. Differ. Equ. (2020), Paper no. 569, 7 pp.
• [4] L. Carlitz Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math. 15 (1979), 51–88.
• [5] D. S. Kim, T. Kim, and D. V. Dolgy, A note on degenerate Bernoulli numbers and polynomials associated with p-adic invariant integral on p , Appl. Math. Comput. 259 (2015), 198–204.
• [6] T. Kim, D. S. Kim, and J.-J. Seo, Fully degenerate poly-Bernoulli numbers and polynomials, Open Math. 14 (2016), no. 1, 545–556.
• [7] D. S. Kim, H. K. Kim, T. Kim, H. Lee, and S. Park, Multi-Lah numbers and multi-Stirling numbers of the first kind, Adv. Differ. Equ. 2021 (2021), Paper no. 411, 9 pp.
• [8] T. Kim, D. S. Kim, H. Lee, and J.-W. Park, A note on degenerate r -Stirling numbers, J. Inequal. Appl. 2020 (2020), Paper no. 225, 12 pp.
• [9] T. Kim, Y. Yao, D. S. Kim, and G.-W. Jang, Degenerate r -Stirling numbers and r -Bell polynomials, Russ. J. Math. Phys. 25 (2018), no. 1, 44–58.
• [10] T. Kim, D. S. Kim, H. K. Kim, and H. Lee, Some properties on degenerate Fubini polynomials, Appl. Math. Sci. Eng. 30 (2022), no. 1, 235–248.
• [11] T. Kim, D. S. Kim, H. Lee, and J. Kwon, On degenerate generalized Fubini polynomials, AIMS Math. 7 (2022), no. 7, 12227–12240.
• [12] D. S. Kim, G.-W. Jang, H.-I. Kwon, and T. Kim, Two variable higher-order degenerate Fubini polynomials, Proc. Jangjeon Math. Soc. 21 (2018), no. 1, 5–22.
• [13] T. Kim, D. S. Kim, L.-C. Jang, H. Lee, and H. Kim, Representations of degenerate Hermite polynomials, Adv. in Appl. Math. 139 (2022), Paper No. 102359.
• [14] T. Kim and D. S. Kim, On some degenerate differential and degenerate difference operators, Russ. J. Math. Phys. 29 (2022), no. 1, 37–46.
• [15] T. Kim, D. S. Kim, H.-Y. Kim, H. Lee, and L.-C. Jang, Degenerate poly-Bernoulli polynomials arising from degenerate polylogarithm, Adv. Differ. Equ. (2020), Paper no. 444, 9 pp.
• [16] S. Araci, A new class of Bernoulli polynomials attached to polyexponential functions and related identities, Adv. Stud. Contemp. Math. (Kyungshang) 31 (2021), no. 2, 195–204.
• [17] A. Bayad and J. Chikhi, Non linear recurrences for Apostol-Bernoulli-Euler numbers of higher order, Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 1, 1–6.
• [18] L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions. Revised and enlarged edition, Reidel Publishing Co., Dordrecht, 1974, xi+343 pp.
• [19] H. W. Gould, Explicit Formulas for Bernoulli Numbers, Amer. Math. Monthly 79 (1972), 44–51.
• [20] H. Haroon and W. A. Khan, Degenerate Bernoulli numbers and polynomials associated with degenerate Hermite polynomials, Commun. Korean Math. Soc. 33 (2018), no. 2, 651–669.
• [21] T. Kim, D. S. Kim, and H. Lee, Some identities involving degenerate r-Stirling numbers, Proc. Jangjeon Math. Soc. 25 (2022), no. 2, 245–252.
• [22] H. K. Kim and D. S. Lee, A new type of degenerate poly-type 2 Euler polynomials and degenerate unipoly-type 2 Euler polynomials, Proc. Jangjeon Math. Soc. 24 (2021), no. 2, 205–222.
• [23] S. K. Sharma, W. A. Khan, S. Araci, and S. S. Ahmed, New construction of type 2 degenerate central Fubini polynomials with their certain properties, Adv. Differ. Equ. 2020 (2020), Paper no. 587, 11 pp.

Uwagi

PL

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).