Analytic Study of $p$-Bessel Functions: Fractional Calculus, Integral Representations, and Complex Extensions

We present a systematic analytic study of the $p$-Bessel functions $\mathcal{J}_{\omega,\varphi}^{[p]}$, a novel class of generalized Bessel functions arising from Fourier analysis on planar domains bounded by $p$-circles, including astroid-type shapes with $0<p\le2$ satisfying $(2/p)\in\mathbb{N}$. While previous work established Hardy-type oscillatory identities for these domains, expressing lattice point discrepancies via $p$-Bessel functions, the present paper focuses on the intrinsic analytic properties of the functions themselves. In particular, we (i) construct a hierarchical structure of $\{\mathcal{J}_{\omega,\varphi}^{[p]}\}_{\omega\ge0}$ using Erd\'{e}lyi-Kober-type fractional derivatives, (ii) derive explicit real-analytic integral representations suitable for investigating axis-dependent asymptotic behavior, and (iii) extend the functions to the complex domain through Poisson-type integral formulas. These results establish $p$-Bessel functions as genuinely new oscillatory kernels, providing a rigorous framework for studying anisotropic oscillatory phenomena and laying the analytic foundation for applications in $p$-circle lattice point problems.