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

Mohamed S. Algolam, Abdulghani Muhyi, Muntasir Suhail (), Neama Haron, Khaled Aldwoah (), W. Eltayeb Ahmed and Amer Alsulami
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Mohamed S. Algolam: Department of Mathematics, College of Science, University of Ha’il, Ha’il 55473, Saudi Arabia
Abdulghani Muhyi: Department of Mechatronics Engineering, Faculty of Engineering and Smart Computing, Modern Specialized University, Sana’a, Yemen
Muntasir Suhail: Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
Neama Haron: Department of Basic Sciences, University College of Haqel, University of Tabuk, Tabuk 71491, Saudi Arabia
Khaled Aldwoah: Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
W. Eltayeb Ahmed: Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi Arabia
Amer Alsulami: Department of Mathematics, Turabah University College, Taif University, Taif 21944, Saudi Arabia

Mathematics, 2025, vol. 13, issue 16, 1-22

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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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