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Volume 13
Issue 16
10.3390/math13162560
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

A Family of q-General Bell Polynomials: Construction, Properties and Applications

by
Mohamed S. Algolam
1,
Abdulghani Muhyi
2,
Muntasir Suhail
3,*,
Neama Haron
4,
Khaled Aldwoah
5,*,
W. Eltayeb Ahmed
6 and
Amer Alsulami
7
1
Department of Mathematics, College of Science, University of Ha’il, Ha’il 55473, Saudi Arabia
2
Department of Mechatronics Engineering, Faculty of Engineering and Smart Computing, Modern Specialized University, Sana’a, Yemen
3
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
4
Department of Basic Sciences, University College of Haqel, University of Tabuk, Tabuk 71491, Saudi Arabia
5
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
6
Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi Arabia
7
Department of Mathematics, Turabah University College, Taif University, Taif 21944, Saudi Arabia
*
Authors to whom correspondence should be addressed.
Mathematics 2025, 13(16), 2560; https://doi.org/10.3390/math13162560
Submission received: 2 July 2025 / Revised: 1 August 2025 / Accepted: 7 August 2025 / Published: 10 August 2025
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
Versions Notes

Abstract

This paper introduces a new family of q-special polynomials, termed q-general Bell polynomials, and systematically explores their structural and analytical properties. We establish their generating functions, derive explicit series representations, and develop recurrence relations to characterize their combinatorial behavior. Additionally, we characterize their quasi-monomial properties and construct associated differential equations governing these polynomials. To demonstrate the versatility and applicability of this family, we investigate certain examples, including the q-Gould–Hopper–Bell and q-truncated exponential-Bell polynomials, deriving analogous results for each. Further, we employ computational tools in Mathematica to examine zero distributions and produce visualizations, offering numerical and graphical insights into polynomial behavior.
Keywords: q-special polynomials; q-Bell polynomials; q-recurrence relations; differential equations; generating function; zero distributions q-special polynomials; q-Bell polynomials; q-recurrence relations; differential equations; generating function; zero distributions

Share and Cite

MDPI and ACS Style

Algolam, M.S.; Muhyi, A.; Suhail, M.; Haron, N.; Aldwoah, K.; Ahmed, W.E.; Alsulami, A. A Family of q-General Bell Polynomials: Construction, Properties and Applications. Mathematics 2025, 13, 2560. https://doi.org/10.3390/math13162560

AMA Style

Algolam MS, Muhyi A, Suhail M, Haron N, Aldwoah K, Ahmed WE, Alsulami A. A Family of q-General Bell Polynomials: Construction, Properties and Applications. Mathematics. 2025; 13(16):2560. https://doi.org/10.3390/math13162560

Chicago/Turabian Style

Algolam, Mohamed S., Abdulghani Muhyi, Muntasir Suhail, Neama Haron, Khaled Aldwoah, W. Eltayeb Ahmed, and Amer Alsulami. 2025. "A Family of q-General Bell Polynomials: Construction, Properties and Applications" Mathematics 13, no. 16: 2560. https://doi.org/10.3390/math13162560

APA Style

Algolam, M. S., Muhyi, A., Suhail, M., Haron, N., Aldwoah, K., Ahmed, W. E., & Alsulami, A. (2025). A Family of q-General Bell Polynomials: Construction, Properties and Applications. Mathematics, 13(16), 2560. https://doi.org/10.3390/math13162560

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