Article ID: MTJPAM-D-23-00006

Title: Study on the degenerate higher-order new Fubini-type numbers and polynomials


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-23-00006; Volume 6 / Issue 3 / Year 2024, Pages 16-33

Document Type: Research Paper

Author(s): ‪Hye Kyung Kim a

aDepartment of Mathematics Education, Daegu Catholic University, Gyeongsan 38430, Republic of Korea

Received: 7 April 2023, Accepted: 31 May 2023, Published: 10 July 2023

Corresponding Author: Hye Kyung Kim (Email address: hkkim@cu.ac.kr)

Full Text: PDF

Abstract

In this paper, we first consider the new two-variables Fubini-type numbers and polynomials of the order r arising from the fermionic p-adic integral on \underbrace{\mathbb{Z}_p \times \cdots \times \mathbb{Z}_p}_{r-times} and those degenerate version. We derive some interesting properties and recurrence relations on those numbers and polynomials. Second, we study the degenerate new two-variables Fubini-type numbers and polynomials of order r as one of the generalizations of these new two-variables Fubini-type numbers and polynomials by using the \lambda-Sheffer sequences. When \lambda \rightarrow 0, we obtain interesting identities for those degenerate new two-variables Fubini-type numbers and polynomials of order r. Some of them include the degenerate and other special polynomials and numbers such as the degenerate Bernoulli polynomials and numbers of order s, the degenerate Euler polynomials and numbers of order s, the degenerate Frobenius-Euler polynomials of order r, the degenerate Lah-Bell polynomials, and so on.

Keywords: Stirling numbers of the second kind, Bell numbers and polynomials, p-adic fermionic integral, two-variables Fubini numbers and polynomials, \lambda-Sheffer sequences

References:
  1. L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Util. Math. 15, 51–88, 1979.
  2. A. Cayley, On the analytical forms called trees-Part II, Philos. Mag. 18, 374–378, 1859.
  3. L. Comtet, Advanced combinatorics: The art of finite and infinite expansions, Dordrecht-Holland, Boston, D. Reidel Publishing Company, 1974. (Translated from the French by J. W. Nienhuys).
  4. G. Dattoli, M. Migliorati and H. M. Srivastava, Sheffer polynomials,monomiality principle, algebraic methods and the theory of classical polynomials, Math. Comput. Model 45, 1033–1041, 2007.
  5. R. Dere, Y. Simsek and H. M. Srivastava, A unified presentation of three families of generalized Apostol type polynomials based upon the theory of the umbral calculus and the umbral algebra, J. Number Theory 133, 3245–3263, 2013.
  6. G. B. Djordjevic and G. V. Milovanovic, Special classes of polynomials, University of Nis, Faculty of Technology, Leskovac, 2014.
  7. O. Herscovici and T. Mansour, Identities involving Touchard polynomials derived from umbral calculus, Adv. Stud. Contemp. Math. (Kyungshang) 25 (1), 39–46, 2015.
  8. N. Kilar and Y. Simsek, A new family of Fubini type numbers and polynomials associated with Apostol-Bernoulli numbers and polynomials, J. Korean Math. Soc. 54 (5), 1605–1621, 2017.
  9. N. Kilar and Y. Simsek, Identities and relations for Fubini type numbers and polynomials via generating functions and p-adic integral approach, Publ. Inst. Math. (Beograd) (N.S.) 106 (120), 113–123, 2019.
  10. D. S. Kim and T. Kim, Degenerate Sheffer sequence and λ-Sheffer sequence, J. Math. Anal. Appl. 23, 2020; Article ID: 124521.
  11. D. S. Kim, T. Kim, H.-I. Kwon and J.-W. Park, Two variable higher-order Fubini polynomials, J. Korean Math. Soc. 55 (4), 975–986, 2018.
  12. H. K. Kim, Degenerate Lah-Bell polynomials arising from degenerate Sheffer sequences, Adv. Difference Equ. 2020, 2020; Article ID: 687.
  13. H. K. Kim and T. Kim, The new Fubini-type numbers and polynomials, DOI: 10.13140/RG.2.2.23656.34566
  14. T. Kim, A note on degenerate Stirling polynomials of the second kind, Proc. Jangjeon Math. Soc. 20 (3), 319–331, 2017.
  15. T. Kim, A note on q-Volkenborn itegration, Proc. Jangjeon Math. Soc. 8 (1), 13–17, 2005.
  16. T. Kim, On the analogs of Euler numbers and polynomials associated with p-adic q-integral on ℤp at q = −1, J. Math. Anal. Appl. 331, 779–792, 2007.
  17. T. Kim, D. S. Kim and D. V. Dolgy, On partially degenerate Bell numbers and polynomials, Proc. Jangjeon Math. Soc. 20 (3), 337–345, 2017.
  18. T. Kim, D. S. Kim, H. Y. Kim and H. Lee, Some properties on degenerate Fubini polynomials, Appl. Math. Sci. Eng. 30 (1), 235–248, 2022; https://doi.org/10.1080/27690911.2022.2056169.
  19. T. Kim, D. S. Kim, T. Mansour, S. H. Rim and M. Schork, Umbral calculus and Sheffer sequences of polynomials, J. Math. Phys. 54 (8), 2013; Article ID: 083504; DOI:10.1063/1.4817853.
  20. I. Kucukoglu, Derivative formulas related to unification of generating functions for Sheffer type sequences, AIP Conference Proceedings 2116, 2019; Article ID: 100016; https://doi.org/10.1063/1.5114092.
  21. H. Niederhausen, Rota’s umbral calculus and recursions, Algbra Universalis 49 (4), 435–457, 2003.
  22. S. Roman, The Umbral Calculus, Academic Press, New York, 1984.
  23. S. Roman and G. Rota, The umbral calculus, Adv. Math. 27, 95–188, 1978.
  24. K. Shiratani and S. Yokoyama, An application of p-adic convolutions, Mem. Fac. Sci. Kyushu Univ. Ser. A. 36 (1), 73–83, 1982.
  25. Y. Simsek, Special numbers and polynomials including their generating functions in umbral analysis methods, Axioms 22 (7), 2018; DOI:10.3390/axioms7020022
  26. Y. Simsek, Explicit formulas for p-adic integrals: approach to p-adic distributions and some families of special numbers and polynomials, Montes Taurus J. Pure Appl. Math. 1 (1), 1–76, 2019.
  27. D. Su and Y. He, Some Identities for the two variable Fubini polynomials, Mathematics 7 (2), 2019; Article ID: 115.
  28. V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-adic analysis and mathematical physics, World Scientific, Singapore, 1994.