Home⇛Recoletos Multidisciplinary Research Journal⇛vol. 13 no. 1 (2025)

On Type 2 Degenerate Poly-Frobenius-Euler Polynomials

Roberto B. Corcino | Cristina B Corcino | Waseem A. Khan

Discipline: Mathematics

 

Abstract:

Background: This paper introduces a class of special polynomials called Type 2 degenerate poly-Frobenius-Euler polynomials, defined using the polyexponential function. Motivated by the expanding theory of degenerate versions of classical polynomials, the paper seeks to enrich the mathematical landscape by constructing generalized structures with deeper combinatorial and analytic properties. Methods: The study employs the method of generating functions combined with Cauchy's rule for the product of two series to derive explicit formulas and identities, enabling systematic manipulation of series expansions. From an analytic perspective, the authors utilized the comparison test and principles of uniform convergence to establish that certain integral representations correspond to holomorphic functions. Results: The researchers successfully derived explicit formulas and identities for the Type 2 degenerate poly-Frobenius-Euler polynomials. They established meaningful connections with the degenerate Stirling numbers of the first and second kinds. Furthermore, they introduced the Type 2 degenerate unipolypoly-Frobenius-Euler polynomials, defined via the unipoly function, and thoroughly investigated their various properties, including behaviors under differentiation and integration. Conclusion: The study significantly advances the theory of degenerate polynomials by constructing new polynomial families, derivation of explicit identities, and establishing analytic properties. It opens new avenues for future research by bridging classical and generalized combinatorial sequences within a robust analytic framework.



References:

  1. Alam, N., Khan, W. A., Kızılateş, C., Obeidat, S., Ryoo, C. S., & Diab, N. S. (2023). Some explicit properties of Frobenius–Euler–Genocchi polynomials with applications in computer modeling. Symmetry, 15(7), 1358–1358. https://doi.org/10.3390/sym15071358
  2. Araci, S., & Acikgoz, M. (2012). A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials. Advanced Studies in Contemporary Mathematics, 22(3). Arxiv. https://doi.org/10.48550/arxiv.1205.6590
  3. Carlitz, L. (1956). A degenerate Staudt-Clausen theorem, Arch. Math. (Basel), 7, 28-33.
  4. Carlitz, L. (1979). Degenerate Stirling, Bernoulli and Eulerian numbers. Utilitas Math, 15, 51–88. https://combinatorialpress.com/um/vol15/
  5. Eastham, M. S. P. (1964). On polylogarithms. Proceedings of the Glasgow Mathematical Association, 6(4), 169–171. https://doi.org/10.1017/s2040618500034961
  6. Elizalde, S. L., & Patan, R. A. (2022). Deriving a formula in solving reverse Fibonacci means. Recoletos Multidisciplinary Research Journal, 10(2), 41–45. https://doi.org/10.32871/rmrj2210.02.03
  7. Guan, H., Khan, W. A., & Kızılateş, C. (2023). On generalized bivariate (p,q)-Bernoulli–Fibonacci polynomials and generalized bivariate (p,q)-Bernoulli–Lucas polynomials. Symmetry, 15(4), 943–943. https://doi.org/10.3390/sym15040943
  8. Janson, S. (2013). Euler-Frobenius numbers and rounding. Arxiv; Cornell University. https://arxiv.org/abs/1305.3512
  9. Kaneko, M. (2025). Poly-Bernoulli numbers. Journal de Théorie Des Nombres de Bordeaux, 9(1), 221–228. http://www.numdam.org/item/JTNB_1997__9_1_221_0/
  10. Khan, W. A. (2018). A new class of degenerate Frobenius-Euler-Hermite polynomials. Advanced Studies in Contemporary Mathematics, 28(4), 567–576.
  11. Khan, W. A. (2023). Construction of the type 2 degenerate poly-Euler polynomials and numbers. Matematički Vesnik, 75(3), 147–156. https://doi.org/10.57016/MV-nr25PM82
  12. Khan, W. A., Alatawi, M. S., & Duran, U. (2023). Applications and Properties for Bivariate Bell-Based Frobenius-Type Eulerian Polynomials. Journal of Function Spaces, 2023. https://doi.org/10.1155/2023/5205867
  13. Khan, W. A., Duran, U., Younis, J., & Ryoo, C. S. (2023). On some extensions for degenerate Frobenius-Euler-Genocchi polynomials with applications in computer modeling. Applied Mathematics in Science and Engineering, 32(1). https://doi.org/10.1080/27690911.2023.2297072
  14. Khan, W. A., & Kamarujjama, M. (2023). A note on type 2 poly-Bernoulli polynomials of the second kind. Palestine Journal of Mathematics, 12(3), 74–84. http://pjm.ppu.edu/sites/default/files/papers/PJM_12%283%29_2023_74_to_84.pdf
  15. Kim, D. S., & Kim, T. (2013). A note on higher-order Bernoulli polynomials. Journal of Inequalities and Applications, 2013. https://doi.org/10.1186/1029-242x-2013-111
  16. Kim, D. S., & Kim, T. (2019). A note on polyexponential and unipoly functions. Russian Journal of Mathematical Physics, 26, 40–49. https://doi.org/10.1134/s1061920819010047
  17. Kim, T. (2017). A note on degenerate stirling polynomials of the second kind. Proceedings of the Jangjeon Mathematical Society, 20(3). Arxiv. https://doi.org/10.48550/arxiv.1704.02290
  18. Kim, T., & Kim, D. S. (2018). An identity of symmetry for the degenerate Frobenius-Euler polynomials. Mathematica Slovaca, 68(1), 239–243. https://doi.org/10.1515/ms-2017-0096
  19. Kim, T., & Kim, D. S. (2020). Degenerate polyexponential functions and degenerate Bell polynomials. Journal of Mathematical Analysis and Applications, 487(2), 124017–124017. https://doi.org/10.1016/j.jmaa.2020.124017
  20. Kim, T., Kim, D. S., Kim, H. Y., & Jang, L.-C. (2020). Degenerate poly-Bernoulli numbers and polynomials. Informatica, 31(3).
  21. Kim, T., Kwon, H.-I., & Seo, J.-J. (2015). On the degenerate Frobenius-Euler polynomials. Global Journal of Pure and Applied Mathematics, 11(5). Arxiv. https://doi.org/10.48550/arXiv.1507.04846
  22. Kurt, B. (2017). Poly-Frobenius-Euler polynomials. AIP Conference Proceesings, 1863(1). https://doi.org/10.1063/1.4992468
  23. Kurt, B., & Simsek, Y. (2013). On the generalized Apostol-type Frobenius-Euler polynomials. Advances in Difference Equations, 2013(1). https://doi.org/10.1186/1687-1847-2013-1
  24. Muhiuddin, G., Khan, W. A., & Al-Kadi, D. (2021). Construction on the degenerate Poly-Frobenius-Euler polynomials of complex variable. Journal of Function Spaces, 2021. https://doi.org/10.1155/2021/3115424
  25. Muhiuddin, G., Khan, W. A., & Al-Kadi, D. (2022). Some identities of the degenerate poly-Cauchy and Unipoly Cauchy polynomials of the second kind. Computer Modeling in Engineering & Sciences, 132(3), 763–779. https://doi.org/10.32604/cmes.2022.017272
  26. Muhiuddin, G., Khan, W. A., & Younis, J. (2021). Construction of Type 2 Poly-Changhee Polynomials and Its Applications. Journal of Mathematics, 2021. https://doi.org/10.1155/2021/7167633
  27. Rao, Y., Khan, W. A., Araci, S., & Ryoo, C. S. (2023). Explicit properties of Apostol-Type Frobenius–Euler polynomials involving q-Trigonometric functions with applications in computer modeling. Mathematics, 11(10), 2386. https://doi.org/10.3390/math11102386
  28. Zhang, C., Khan, W. A., & Kızılateş, C. (2023). On (p,q)–Fibonacci and (p,q)–Lucas polynomials associated with Changhee numbers and their properties. Symmetry, 15(4), 851–851. https://doi.org/10.3390/sym15040851

Tools


Copyright © 2026  KITE Digital Educational Solutions |  Exclusively distributed by CE-Logic