Further Results for Hermite-Based Milne-Thomson Type Fubini Polynomials with Trigonometric Functions

Abstract

This paper examines generating functions of r-parametric Hermite-based Milne-Thomson polynomials. Using generating function methods, the relationships among these polynomials, Fubini type polynomials, and trigonometric functions are given. Moreover, new formulas are derived by utilizing not only the generating functions of these polynomials but also associated functional equations. These formulas pertain to r-parametric Hermite-based sine-and cosine-Milne-Thomson Fubini polynomials, as well as Stirling type polynomials and numbers. Additionally, by analyzing special cases of newly obtained results, some known formulas are also derived. Furthermore, some identities involving secant and cosecant numbers are derived through the properties of trigonometric functions. Special polynomials and their generating functions are an important tool for solving some problems in many areas such as combinatorics and number theory. By introducing new formulas, this paper significantly enhances these problems-solving abilities in these areas. Consequently, these results have potential to shed light on important applications in mathematics, engineering, and mathematical physics.

Keywords

• Hermite-Based Milne-Thomson Type Polynomials
• Stirling Type Polynomials and Numbers
• Trigonometric Functions
• Generating Functions

References

1. Agyuz, E. (2024). On convergence properties of Fubini-type polynomials. AIP Conference Proceedings, 3094(090012). https://doi.org/10.1063/5.0210608
2. Ali, M., & Paris, R. B. (2022). Multi-index Fubini-type polynomials. Montes Taurus Journal of Pure and Applied Mathematics, 4(1), 97-106.
3. Bayad, A., & Simsek, Y. (2014). Convolution identities on the Apostol-Hermite base of two variables polynomials. Differential Equations and Dynamical Systems, 22(3), 309-318. https://doi.org/10.1007/s12591-013-0181-7
4. Cesarano, C., Ramírez, W., & Khan, S. (2022). A new class of degenerate Apostol-type Hermite polynomials and applications. Dolomites Research Notes on Approximation, 15(1), 1-10. https://doi.org/10.14658/PUPJ-DRNA-2022-1-1
5. Charalambides, C. A. (2005). Combinatorial methods in discrete distributions. Hoboken: John Wiley & Sons Inc., Publication.
6. Dattoli, G., Chiccoli, C., Lorenzutta, S., Maino, G., & Torre, A. (1994). Theory of generalized Hermite polynomials. Computers & Mathematics with Applications, 28(4), 71-83. https://doi.org/10.1016/0898-1221(94)00128-6
7. Dattoli, G., Lorenzutta, S., Maino, G., Torre, A., & Cesarano, C. (1996). Generalized Hermite polynomials and supergaussian forms. Journal of Mathematical Analysis and Applications, 203, 597-609. https://doi.org/10.1006/jmaa.1996.0399
8. Diagana, T., & Maïga, H. (2017). Some new identities and congruences for Fubini numbers. Journal of Number Theory, 173, 547-569. https://doi.org/10.1016/j.jnt.2016.09.032

9. Fadel, M., Raza, N., & Du, W.-S. (2024). On q-Hermite polynomials with three variables: Recurrence relations, q-differential equations, summation and operational formulas. Symmetry, 16(4), 1-19. https://doi.org/10.3390/sym16040385
10. Kereskényi-Balogh, Z., & Nyul, G. (2021). Fubini numbers and polynomials of graphs. Mediterranean Journal of Mathematics, 18(230), 1-10. https://doi.org/10.1007/s00009-021-01838-x
11. Kilar, N. (2021). Generating Functions of Hermite Type Milne-Thomson Polynomials and Their Applications in Computational Sciences. Antalya: PhD Thesis, University of Akdeniz.
12. Kilar, N. (2023a). On computational formulas for parametric type polynomials and its applications. Journal of Balıkesir University Institute of Science and Technology, 25(1), 13-30. https://doi.org/10.25092/baunfbed.1083754
13. Kilar, N. (2023b). Combinatorial sums and identities associated with functional equations of generating functions for Fubini type polynomials. GUJ Sci, 36(2), 807-817. https://doi.org/10.35378/gujs.989270
14. Kilar, N., & Simsek, Y. (2017). A new family of Fubini numbers and polynomials associated with Apostol-Bernoulli numbers and polynomials. Journal of the Korean Mathematical Society, 54(5), 1605-1621. https://doi.org/10.4134/JKMS.j160597
15. Kilar, N., & Simsek, Y. (2021). Computational formulas and identities for new classes of Hermite-Based Milne-Thomson type polynomials: Analysis of generating functions with Euler’s formula. Mathematical Methods in the Applied Sciences, 44(8), 6731-6762. https://doi.org/10.1002/mma.7220
16. Kim, D. S., & Kim, T. (2018). Some p-adic integrals on Z_p associated with trigonometric functions. Russian Journal of Mathematical Physics, 25(3), 300-308. https://doi.org/10.1134/S1061920818030032
17. Simsek, Y. (2013). Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications. Fixed Point Theory Applications, 2013(87), 1-28. https://doi.org/10.1186/1687-1812-2013-87
18. Simsek, Y. (2024). Formulas for p-adic q-integrals including falling-rising factorials, combinatorial sums and special numbers. RACSAM, 118(92), 1-52. https://doi.org/10.1007/s13398-024-01592-1
19. Srivastava, H. M. (1976). A note on a generating function for the generalized Hermite polynomials. Indagationes Mathematicae (Proceedings), 79(5), 457-461. https://doi.org/10.1016/S1385-7258(76)80009-1
20. Srivastava, H. M., Srivastava, R., Muhyi, A., Yasmin G., Islahi H., & Araci, S. (2021). Construction of a new family of Fubini-type polynomials and its applications. Advances in Difference Equations, 2021(36), 1-25. https://doi.org/10.1186/s13662-020-03202-x
21. Zayed, M., Wani, S. A., Oros, G. I., & Ramŕez, W. (2024). A study on extended form of multivariable Hermite-Apostol type Frobenius-Euler polynomials via fractional operators. AIMS Mathematics, 9(6), 16297-16312. https://doi.org/10.3934/math.2024789