On Bell based Appell polynomials

Zeynep ÖZAT, (Gazi Üniversitesi, Fen Bilimleri Enstitüsü, Ankara, Türkiye)
Mehmet Ali ÖZARSLAN, (Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, via Mersin 10, Gazimağusa, North Cyprus, Turkey)
Bayram ÇEKİM (Gazi Üniversitesi, Fen Fakültesi, Matematik Bölümü, Ankara, Türkiye)
Turkish Journal of Mathematics
0 0
Yıl: 2023 Cilt: 47 Sayı: 4 Sayfa Aralığı: 1099 - 1128 Metin Dili: İngilizce DOI: 10.55730/1300-0098.3415 İndeks Tarihi: 14-03-2024

On Bell based Appell polynomials

Öz:
Recently, several Bell based polynomials such as Bernoulli, Euler, Genocchi and Apostol versions were defined and investigated. The main aim of this paper is to introduce the general family of Bell based Appell polynomials, which includes many new members in addition to the existing ones, and to investigate their properties including determinantal representation, recurrence relation, derivative formula, addition formula, shift operators and differential equation. Furthermore, we introduce 2-iterated Bell-Appell polynomials and investigate their similar properties. With the help of this 2-iterated family, we also obtain the closed form summation formulae between the usual and the generalized versions of the Bell based Appell polynomials. Finally, we introduce the Bell based Bernoulli-Euler, Bernoulli-Genocchi, Euler-Genocchi and Stirling-Appell polynomials of the second kind as special cases of 2-iterated Bell based Appell polynomials and state the corresponding results.
Anahtar Kelime: Appell polynomials Bell polynomials Bell based Bernoulli polynomials Bell based Euler polynomials Bell based Genocchi polynomials Bell based Stirling polynomials of the second kind

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık
  • [1] Alam N, Khan WA, Ryoo CS. A note on Bell-based Apostol-type Frobenius-Euler polynomials of complex variable with its certain applications. Mathematics 2022; 10 (12): 1-26. https://doi.org/10.3390/math10122109
  • [2] Appell P. Sur une classe de pôlynomes. Annales Scientifiques de l’Ecole Normale Supérieure 1980; 9: 119-144 (in French).
  • [3] Bretti G, Natalini P, Ricci PE. Generalizations of the Bernoulli and Appell polynomials. Abstract and Applied Analysis 2004; 2004 (7): 613-623.
  • [4] Bretti G, Ricci PE. Multidimensional extension of the Bernoulli and Appell polynomials. Taiwanese Journal of Mathematics 2004; 8 (3): 415-428. https://doi.org/10.11650/twjm/1500407662
  • [5] Carlitz L. Bernoulli numbers. Fibonacci Quarterly 1968; 6 (3): 71-85.
  • [6] Carlitz L. Some remarks on the Bell numbers. Fibonacci Quarterly 1980; 18 (1): 66-73.
  • [7] Comtet L. Advanced Combinatorics: The Art of Finite and Infinite Expansions. Dordrecht, Holland: D. Reidel Publishing Company, 1974.
  • [8] Costabile FA, Longo E. ∆h -Appell sequences and related interpolation problem. Numerical Algorithms 2013; 63: 165-186. https://doi.org/ 10.1007/s11075-012-9619-1
  • [9] Djordjević GB, Milovanović GV. Special Classes of Polynomials. Leskovac: University of Nis, Faculty of Technology, 2014.
  • [10] Duran U, Acikgoz M. Bell-based Genocchi polynomials. New Trends in Mathematical Sciences 2021; 9 (1): 50-55. https://doi.org/10.20852/ntmsci.2021.428
  • [11] Duran U, Araci S, Acikgoz M. Bell-based Bernoulli polynomials with applications. Axioms 2021; 10 (1): 29. https://doi.org/10.3390/axioms10010029
  • [12] He MX, Ricci PE. Differential equation of Appell polynomials via the factorization method. Journal of Computa- tional and Applied Mathematics 2002; 139 (2): 231-237. https://doi.org/10.1016/S0377-0427(01)00423-X
  • [13] Howard FT. Bell polynomials and degenerate Stirling numbers. Rendiconti del Seminario Matematico della Uni- versità di Padova 1979; 61: 203-219.
  • [14] Kamarujima M, Husain, S. Bell based Apostol-Bernoulli polynomials and its properties. International Journal of Applied and Computational Mathematics 2022; 8 (18): 1-12. https://doi.org/10.1007/s40819-021-01213-0
  • [15] Khan N, Husain S. Analysis of Bell based Euler polynomials and their application. International Journal of Applied and Computational Mathematics 2021; 7 (5): 1-16. https://doi.org/10.1007/s40819-021-01127-x
  • [16] Khan N, Husain S. Bell based Apostol type polynomials and its properties. arXiv:2109.10550 [math.NT]. https://doi.org/10.48550/arXiv.2109.10550
  • [17] Khan S, Ali M. A linear algebra approach to the hybrid Sheffer-Appell polynomials. Mathematical Sciences 2019; 13 (2): 153-164. https://doi.org/10.1007/s40096-019-0286-4
  • [18] Khan S, Al-Saad MWM, Khan R. Laguerre-based Appell polynomials: properties and applications. Mathematical and Computer Modelling 2010; 52 (1-2): 247-259. https://doi.org/10.1016/j.mcm.2010.02.022
  • [19] Khan S, Nahid T. Finding non-linear differential equations and certain identities for the Bernoulli–Euler and Bernoulli–Genocchi numbers.SN Applied Sciences 2019; 1(217): 1-9. https://doi.org/10.1007/s42452-019-0222-0
  • [20] Khan S, Raza N. General-Appell polynomials within the context of monomiality principle. International Journal of Analysis 2013; 1-11. https://doi.org/ 10.1155/2013/328032
  • [21] Khan S, Raza N. 2-iterated Appell polynomials and related numbers. Applied Mathematics and Computation 2013; 219 (17): 9469-9483. https://doi.org/10.1016/j.amc.2013.03.082
  • [22] Khan S, Riyaset M. Determinantal approach to certain mixed special polynomials related to Gould–Hopper poly- nomials. Applied Mathematics and Computation 2015; 251: 599-614. https://doi.org/10.1016/j.amc.2014.11.081
  • [23] Khan S, Riyaset M. Differential and integral equations for the 2-iterated Bernoulli, 2-iterated Euler and Bernoulli- Euler polynomials. Georgian Mathematical Journal 2020; 27: 375-389. https://doi.org/10.1515/gmj-2018-0062
  • [24] Khan S, Yasmin G, Ahmad N. A note on truncated exponential based Appell polynomials. Bulletin of the Malaysian Mathematical Sciences Society 2017; 40: 373-388. https://doi.org/10.1007/s40840-016-0343-1
  • [25] Khan S, Yasmin G, Khan R, Hassan NAM. Hermite-based Appell polynomials: properties and applications. Journal of Mathematical Analysis and Applications 2009; 351 (2): 756-764. https://doi.org/10.1016/j.jmaa.2008.11.002
  • [26] Kim T, Kim DS, Jang GW. On central complete and incomplete Bell polynomials I. Symmetry 2019; 11 (2): 1-12. https://doi.org/10.3390/sym11020288
  • [27] Kim T, Kim DS, Jang LC, Lee H, Kim H. Representations of degenerate Hermite polynomials. Advances in Applied Mathematics 2022; 139: 1-18. https://doi.org/ 10.1016/j.aam.2022.102359
  • [28] Nahid T, Khan S. Construction of some hybrid relatives of Laguerre-Appell polynomials associated with Gould- Hopper matrix polynomials. The Journal of Analysis 2021; 29 (3): 927-946. https://doi.org/10.1007/s41478-020- 00288-0
  • [29] Nahid T, Khan S. Differential equations for certain hybrid special matrix polynomials. Boletim da Sociedade Paranaense de Matemática 2022; 41 (2023): 1-10. https://doi.org/10.5269/bspm.52758
  • [30] Özarslan MA. Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials. Advances in Difference Equations 2013; 116: 1-13. https://doi.org/10.1186/1687-1847-2013-116
  • [31] Özarslan MA. Unified Apostol-Bernoulli, Euler and Genocchi polynomials. Computers and Mathematics with Applications 2011; 62 (6): 2452-2462. https://doi.org/10.1016/j.camwa.2011.07.031
  • [32] Özarslan MA, Yaşar BY. ∆h -Gould-Hopper Appell polynomials, Acta Mathematica Scientia 2021; 41B (4): 1196- 1222. https://doi.org/10.1007/s10473-021-0411-y
  • [33] Özarslan MA, Yılmaz B. A set of finite order differential equations for the Appell polynomials. Journal of Compu- tational Applied Mathematics 2014; 259: 108-116. https://doi.org/10.1016/j.cam.2013.08.006
  • [34] Riyasat M, Nahid T, Khan S. An algebraic approach to degenerate Appell polynomials and their hybrid forms via determinants. Acta Mathematica Scientia 2022; 43B: 1-17. https://doi.org/10.1007/s10473-023-0215-3
  • [35] Roman S. The Umbral Calculus. Pure and Applied Mathematics. New York, NY, USA: Academic Press, Inc., 1984.
  • [36] Sheffer IM. Note on Appell polynomials. Bulletin of the American Mathematical Society 1945; 51 (10): 739-744.
  • [37] Srivastava HM, Choi J. Zeta and q -Zeta Functions and Associated Series and Integrals. New York, NY, USA: Elsevier Science Publishers, 2012.
  • [38] Srivastava HM, Özarslan MA, Yaşar BY. Difference equations for a class of twice iterated ∆h -Appell sequences of polynomials. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 2019; 113: 1851-1871. https://doi.org/10.1007/s13398-018-0582-0
  • [39] Srivastava HM, Özarslan MA, Yılmaz B. Some families of differential equations associated with the Hermite- based Appell polynomials and other classes of Hermite-based polynomials. Filomat 2014; 28 (4): 695-708. https://doi.org/10.2298/FIL1404695S
  • [40] Srivastava HM, Yasmin G, Muhyi A, Araci, S. Certain results for the twice-iterated 2 D q -Appell polynomials. Symmetry 2019; 11 (10): 1307. https://doi.org/ 10.3390/sym11101307
  • [41] Varma S, Yaşar BY, Özarslan MA. Hahn-Appell polynomials and their d -orthogonality. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 2019; 113 (3): 2127-2143. https://doi.org/10.1007/s13398-018-0607-8
  • [42] Wani SA, Nahid T. Finding determinant and integral forms of the 2 -iterated 2D Appell polynomials. The Journal of Analysis 2022; 30: 731-747. https://doi.org/10.1007/s41478-021-00369-8
APA ÖZAT Z, Özarslan M, Çekim B (2023). On Bell based Appell polynomials. , 1099 - 1128. 10.55730/1300-0098.3415
Chicago ÖZAT Zeynep,Özarslan Mehmet Ali,Çekim Bayram On Bell based Appell polynomials. (2023): 1099 - 1128. 10.55730/1300-0098.3415
MLA ÖZAT Zeynep,Özarslan Mehmet Ali,Çekim Bayram On Bell based Appell polynomials. , 2023, ss.1099 - 1128. 10.55730/1300-0098.3415
AMA ÖZAT Z,Özarslan M,Çekim B On Bell based Appell polynomials. . 2023; 1099 - 1128. 10.55730/1300-0098.3415
Vancouver ÖZAT Z,Özarslan M,Çekim B On Bell based Appell polynomials. . 2023; 1099 - 1128. 10.55730/1300-0098.3415
IEEE ÖZAT Z,Özarslan M,Çekim B "On Bell based Appell polynomials." , ss.1099 - 1128, 2023. 10.55730/1300-0098.3415
ISNAD ÖZAT, Zeynep vd. "On Bell based Appell polynomials". (2023), 1099-1128. https://doi.org/10.55730/1300-0098.3415
APA ÖZAT Z, Özarslan M, Çekim B (2023). On Bell based Appell polynomials. Turkish Journal of Mathematics, 47(4), 1099 - 1128. 10.55730/1300-0098.3415
Chicago ÖZAT Zeynep,Özarslan Mehmet Ali,Çekim Bayram On Bell based Appell polynomials. Turkish Journal of Mathematics 47, no.4 (2023): 1099 - 1128. 10.55730/1300-0098.3415
MLA ÖZAT Zeynep,Özarslan Mehmet Ali,Çekim Bayram On Bell based Appell polynomials. Turkish Journal of Mathematics, vol.47, no.4, 2023, ss.1099 - 1128. 10.55730/1300-0098.3415
AMA ÖZAT Z,Özarslan M,Çekim B On Bell based Appell polynomials. Turkish Journal of Mathematics. 2023; 47(4): 1099 - 1128. 10.55730/1300-0098.3415
Vancouver ÖZAT Z,Özarslan M,Çekim B On Bell based Appell polynomials. Turkish Journal of Mathematics. 2023; 47(4): 1099 - 1128. 10.55730/1300-0098.3415
IEEE ÖZAT Z,Özarslan M,Çekim B "On Bell based Appell polynomials." Turkish Journal of Mathematics, 47, ss.1099 - 1128, 2023. 10.55730/1300-0098.3415
ISNAD ÖZAT, Zeynep vd. "On Bell based Appell polynomials". Turkish Journal of Mathematics 47/4 (2023), 1099-1128. https://doi.org/10.55730/1300-0098.3415
Sayfa Oluşturulma Tarihi
24-04-2024 7:51

İLETİŞİM

TÜBİTAK ULAKBİM TR Dizin

Yüzüncüyıl, İşçi Blokları Mahallesi

Muhsin Yazıcıoğlu Caddesi No:51/C

06530 Çankaya / ANKARA +90 (312) 298 92 00

trdizin@tubitak.gov.tr

TÜBİTAK ULAKBİM Ulusal Akademik Ağ ve Bilgi Merkezi Cahit Arf Bilgi Merkezi © Tüm Hakları Saklıdır. Gizlilik Politikası Kullanım Şartları