Analysis of a New Sheffer Polynomial Family with Properties and Applications

This paper introduces a novel family of polynomials ₍^ (x), defined via the exponential generating function e^ t+x (1-e^{t) }. These polynomials generalize and unify several classical families, including Bell and Touchard polynomials, through a Sheffer-type structure. We derive explicit expressions, establish recurrence relations, and explore algebraic properties that reveal their rich combinatorial significance. Applications are discussed across diverse domains such as combinatorics, differential equations and mathematical physics and statistical mechanics. The versatility of the ₍^ (x) polynomials suggests promising directions for future research, including orthogonality analysis, asymptotic behavior, q -analogues, multivariate generalizations and applications in stochastic modeling and algorithmic frameworks.