Analytic normalization and geometric behavior of generalized Bessel functions

This work introduced normalized representations of generalized Bessel functions, denoted by ${}_k{H}_{\beta ,g}(\mathrm{\wp })$ and ${}_k{M}_{\beta ,g}(\mathrm{\wp };s)$ in terms of $k$ and $(s,k)$, where $s>k>0$ and $s<1$. The motivation stemed from the increasing relevance of such functions in diverse fields such as mathematical physics, wave dynamics and fractional calculus, where classical Bessel functions may be limited. This study also gave detailed geometric investigation of these generalized Bessel functions. We derived sufficient conditions for these functions to be starlike and convex of order $\zeta$ in the open unit disk $\mathbb{U}$. The results of our analysis revealed enhanced structural flexibility over classical counterparts, making them well-suited for advanced applications in complex modeling theory. Several illustrative examples and plots were provided to validate and visualize the behavior of the proposed functions, highlighting their potential in mathematical modeling and theoretical studies. These findings enhanced the geometric function theory framework for special functions and provided a solid foundation for future analytical and applied investigations.