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

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.