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

TL;DR

This paper introduces a new class of generalized Bessel functions called $p$-Bessel functions, exploring their fractional calculus, integral representations, and complex extensions to analyze anisotropic oscillations and lattice point problems.

Contribution

It develops a hierarchical fractional derivative structure, explicit integral representations, and complex extensions for $p$-Bessel functions, advancing their theoretical understanding.

Findings

Constructed a hierarchical structure of $p$-Bessel functions using fractional derivatives.

Derived explicit integral representations for asymptotic analysis.

Extended $p$-Bessel functions to the complex domain via Poisson-type formulas.

Abstract

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.