An Analytical Study of Two-Dimensional Bell Polynomials and Their Properties

This study investigates two-dimensional Bell polynomials, emphasizing their fundamental properties and applications in mathematical analysis. Utilizing the framework of generating functions, we derive explicit representations, summation formulae, recurrence relations, and addition formulas for these polynomials. Additionally, we present their matrix form and product formula, further enriching their structural analysis. Furthermore, we introduce the 2D Bell-based Stirling polynomials of the second kind and explore their associated properties. This research aims to enhance the theoretical understanding of Bell polynomials and their broader applications in mathematical analysis.