Some applications of partial r-Bell polynomials

In this work, we introduce and investigate a novel class of polynomials, which we term t-successive exponential partial Bell and r-Bell polynomials. These polynomials generalize the well-studied exponential partial Bell and r-Bell polynomials and are closely related to classical combinatorial sequences such as Stirling numbers, Bell numbers, and their polynomial extensions. Our motivation stems from the unified framework proposed by Mihoubi and Rahmani in(2017), which provides a systematic approach for constructing exponential partial r-Bell polynomials, of which many known combinatorial numbers emerge as special cases.

We begin by establishing explicit formulas for these polynomials and presenting their generating functions, which serve as a fundamental tool for deriving structural properties. Through careful analysis, we deduce several recurrence relations that encapsulate the inherent combinatorial structure of these polynomials. In particular, we provide combinatorial interpretations in terms of set partitions, illustrating how elements may be organized into subsets under specific constraints, thereby extending the classical interpretations associated with Bell and Stirling numbers.

t-successive exponential partial Bell and r-Bell polynomials exhibit rich combinatorial and algebraic properties. They offer a unifying perspective that bridges existing sequences and opens new avenues for exploration in discrete mathematics and combinatorial enumeration. We illustrate their utility through representative examples and demonstrate how their generating functions facilitate the derivation of identities and recurrence relations.

Our study not only generalizes known results but also provides a versatile framework for further investigations. By combining explicit constructions, recurrence relations, and combinatorial interpretations, these polynomials contribute to a deeper understanding of the interplay between classical combinatorial sequences and their extensions. The results presented herein offer potential applications in sequence transformations, partition theory, and the broader study of combinatorial structures.