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

Abstract

We present a systematic analytic study of the $p$-Bessel functions $\mathcal{J}_{ω,\varphi}^{[p]}$, 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 $\{\mathcal{J}_{ω,\varphi}^{[p]}\}_{ω\ge0}$ using Erdélyi-Kober-type fractional derivatives, (ii) derive explicit real-analytic integral representations suitable for axis-dependent asymptotic estimates, 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.

Figures (1)

• Figure 1: Examples of the $p$-circle and the approximation by unit squares.

Theorems & Definitions (18)

• Theorem 1.1: K3, Theorem 1.2; Hardy-type oscillatory identity for the astroid-type $p$-circle
• Theorem 1.2: Erdélyi–Kober-type fractional differential identity
• Theorem 1.3: Integral representation
• Theorem 1.4: Poisson-type integral representation for $p$-Bessel Functions
• Lemma 2.1: K3, (2.7); Order-raising integral formula (analogue of that for $J_{\omega}$)
• Lemma 2.2: K3, Proposition 2.3; Order-lowering differential formula (analogue of that for $J_{\omega}$)
• proof : Proof of Theorem \ref{['E-K-derJ']}
• Remark 2.3
• Proposition 2.4
• proof