On certain properties of Hybrid Sheffer– $\lambda$-type special polynomials and their applications

In this work, a new class of Gould-Hopper Sheffer- $\lambda$ polynomials was introduced by combining the structural features of Gould-Hopper polynomials with the general framework of Sheffer- $\lambda$ sequences. The proposed family was defined through an appropriate exponential generating function involving trigonometric factors and was shown to possess rich algebraic and operational properties. Explicit series representations and determinant forms were derived using Riordan array techniques and Cramer's rule. By employing the monomiality principle, the associated multiplicative and derivative operators were constructed, establishing the quasi-monomial character of the introduced polynomials and leading to the corresponding differential equations. Furthermore, several important subclasses, including Gould-Hopper-Bernoulli- $\lambda$, Euler- $\lambda$, Genocchi- $\lambda$, and Laguerre- $\lambda$ polynomials, were obtained as illustrative examples. These examples demonstrated the unifying nature of the proposed framework and highlighted its potential applicability in operational calculus, special functions, and mathematical physics.