Properties and applications of Bell polynomials of two variables
Authors
M. Zayed
- Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia.
Sh. A. Wani
- Department of Applied Sciences, Symbiosis Institute of Technology, Symbiosis International (Deemed University) (SIU), Pune, Maharashtra, India.
W. Ramirez
- Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Rome, Italy.
- Department of Natural and Exact Sciences, Universidad de la Costa, Calle 58 N 55-66, 080002 Barranquilla, Colombia.
M. A. Alqudah
- School of Basic Sciences and Humanities, German Jordanian University, Amman, 11180, Jordan.
F. F. Gandara
- Department of Natural and Exact Sciences, Universidad de la Costa, Calle 58 N 55-66, 080002 Barranquilla, Colombia.
Abstract
In this article, we introduce Bell polynomials of two variables within the framework of generating functions and explore various properties associated with them. Specifically, we delve into explicit representations, summation formulae, recurrence relations, and addition formulas. Additionally, we present the matrix form and product formula for these polynomials. Finally, we introduce the two-variable Bell-based Stirling polynomials of the second kind and outline their corresponding results. This study contributes to a deeper understanding of the properties and applications of Bell polynomials in mathematical analysis.
Share and Cite
ISRP Style
M. Zayed, Sh. A. Wani, W. Ramirez, M. A. Alqudah, F. F. Gandara, Properties and applications of Bell polynomials of two variables, Journal of Mathematics and Computer Science, 35 (2024), no. 3, 291--303
AMA Style
Zayed M., Wani Sh. A., Ramirez W., Alqudah M. A., Gandara F. F., Properties and applications of Bell polynomials of two variables. J Math Comput SCI-JM. (2024); 35(3):291--303
Chicago/Turabian Style
Zayed, M., Wani, Sh. A., Ramirez, W., Alqudah, M. A., Gandara, F. F.. "Properties and applications of Bell polynomials of two variables." Journal of Mathematics and Computer Science, 35, no. 3 (2024): 291--303
Keywords
- Special polynomials
- monomiality principle
- explicit form
- operational connection
- symmetric identities
- summation formulae
MSC
- 33E20
- 33C45
- 33B10
- 33E30
- 11T23
References
-
[1]
R. Alyusof, S. A. Wani, Certain Properties and Applications of h Hybrid Special Polynomials Associated with Appell Sequences, Fractal Fract., 7 (2023), 1–10
-
[2]
A. Bayad, T. Kim, Identities for Apostol-type Frobenius-Euler polynomials resulting from the study of a nonlinear operator, Russ. J. Math. Phys., 23 (2016), 164–171
-
[3]
D. Bedoya, C. Cesarano, W. Ram´ırez, L. Castilla, A new class of degenerate biparametric Apostol-type polynomials, Dolomites Res. Notes Approx., 16 (2023), 10–19
-
[4]
E. T. Bell, Exponential polynomials, Ann. of Math. (2), 35 (1934), 258–277
-
[5]
R. P. Boas, Jr., R. C. Buck, Polynomial Expansions of Analytic Functions, Springer-Verlag, Berlin-G¨ottingen- Heidelberg (1958)
-
[6]
L. Carlitz, Some remarks on the Bell numbers, Fibonacci Quart., 18 (1980), 66–73
-
[7]
L. Comtet, Advanced combinatorics, D. Reidel Publishing Co., Dordrecht (1974)
-
[8]
G. Dattoli, P. L. Ottaviani, A. Torre, L. V´azquez, Evolution operator equations: integration with algebraic and finitedifference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory, Riv. Nuovo Cimento Soc. Ital. Fis. (4), 20 (1997), 3–133
-
[9]
S. Khan, M. A. Pathan, N. A. M. Hassan, G. Yasmin, Implicit summation formulae for Hermite and related polynomials, J. Math. Anal. Appl., 344 (2008), 408–416
-
[10]
T. Kim, An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic p-adic invariant q-integrals on Zp, Rocky Mountain J. Math., 41 (2011), 239–247
-
[11]
T. Kim, Identities involving Frobenius-Euler polynomials arising from non-linear differential equations, J. Number Theory, 132 (2012), 2854–2865
-
[12]
D. S. Kim, T. Kim, Some new identities of Frobenius-Euler numbers and polynomials, J. Inequal. Appl., 2012 (2012), 10 pages
-
[13]
D. S. Kim, T. Kim, D. V. Dolgy, A note on degenerate Bernoulli numbers and polynomials associated with p-adic invariant integral on Zp, Appl. Math. Comput., 259 (2015), 198–204
-
[14]
T. Kim, B. Lee, Some identities of the Frobenius-Euler polynomials, Abstr. Appl. Anal., 2009 (2009), 7 pages
-
[15]
T. Kim, J. J. Seo, Some identities involving Frobenius-Euler polynomials and numbers, Proc. Jangjeon Math. Soc., 19 (2016), 39–46
-
[16]
Y. Quintana, W. Ram´ırez, A. Urieles, Generalized Apostol-type polynomial matrix and its algebraic properties, Math. Rep. (Bucur.), 21 (2019), 249–264
-
[17]
H. M. Srivastava, J. Choi, Zeta and q-Zeta functions and associated series and integrals, Elsevier, Inc., Amsterdam (2012)
-
[18]
H. M. Srivastava, H. L. Manocha, A Treatise on Generating Functions, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York (1984)
-
[19]
W. Wang, T. Wang, Identities on Bell polynomials and Sheffer sequences, Discrete Math., 309 (2009), 1637–1648
-
[20]
S. A. Wani, Two-iterated degenerate Appell polynomials: properties and applications, Arab J. Basic Appl. Sci., 31 (2024), 83–92
-
[21]
S. A. Wani, K. Abuasbeh, G. I. Oros, S. Trabelsi, Studies on Special Polynomials Involving Degenerate Appell Polynomials and Fractional Derivative, Symmetry, 15 (2023), 1–12
-
[22]
N. Zayed, S. A. Wani, A Study on Generalized Degenerate Form of 2D Appell Polynomials via Fractional Operators, Fractal Fract., 7 (2023), 1–14
-
[23]
Z. Zhang, M. Liu, An extension of generalized Pascal matrix and its algebraic properties, Linear Algebra Appl., 271 (1998), 169–177