This study presents a new, generalized family of special polynomials. These polynomials are designated as the Apostol–Hermite–Kampé de Fériet–Bell–Bernoulli-type polynomials. Based on their generating function, the corresponding series expansions and summation formulas are obtained. Furthermore, determinant representation and several differential and integral representations of these polynomials are derived. The fractional extension of this polynomial family is explored, resulting in the formulation of several associated identities. Finally, the study utilizes Mathematica to deliver zero distributions and graphical representations.

This study explores the stochastic Benjamin-Bona-Mahony (BBM) equation with a beta derivative (BD), thereby incorporating multiplicative noise in the Itô sense. We derive various analytical soliton solutions for these equations utilizing two distinct expansion methods: the $\frac{{\mathcal{G}}^{\mathrm{\prime }}}{{\mathcal{G}}^{\mathrm{\prime }}+\mathcal{G}+\mathcal{A}}$-expansion and the modified $\frac{{\mathcal{G}}^{\mathrm{\prime }}}{{\mathcal{G}}^2}$-expansion techniques, both within the framework of beta derivatives. A fractional multistep transformation is employed to convert the equations into nonlinear forms with respect to an independent variable. After performing an algebraic manipulation, the solutions are trigonometric and hyperbolic trigonometric functions. Our analysis demonstrates that the wave behavior is influenced by the fractional-order derivative in the proposed equations, thus providing deeper insights into the wave composition as the fractional order either increases or decreases. Additionally, we explore the effect of white noise on the propagation of the waves solutions. This study underscores the computational robustness and adaptability of the proposed approach to investigate various phenomena in the physical sciences and engineering.

Rotavirus remains a leading cause of gastroenteritis in children under five in low- and middle-income countries due to waning immunity and incomplete vaccine coverage. To address this, we propose a mathematical model to analyze the transmission dynamics with primary and booster vaccination strategies. The model is formulated using the fractal-fractional derivative in the Caputo-Fabrizio sense, which allows for the incorporation of memory effects and hereditary properties in disease evolution. The population is structured into five compartments, including booster-immunized individuals. We derive the disease-free and endemic equilibrium points and analyze their local stability. The basic reproduction number is computed to determine the threshold conditions for disease persistence. We establish the existence and Hyers-Ulam (H-U) stability of the model, and validate the results through numerical simulations using the Adams-Bashforth method (ABM), confirmed by comparison with Runge-Kutta 4th Order (RK-4) solution plots to assess the booster vaccination impact. The results reveal that booster immunization plays a significant role in reducing the infection burden, thereby highlighting its relevance in public health planning.