Analytic Study of $p$-Bessel Functions: Fractional Calculus, Integral Representations, and Complex Extensions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913084187082752 |
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| author | Kitajima, Masaya |
| author_facet | Kitajima, Masaya |
| contents | We present a systematic analytic study of the $p$-Bessel functions $\mathcal{J}_{ω,φ}^{[p]}$, 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 $\{\mathcal{J}_{ω,φ}^{[p]}\}_{ω\ge0}$ using Erdélyi-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21072 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Analytic Study of $p$-Bessel Functions: Fractional Calculus, Integral Representations, and Complex Extensions Kitajima, Masaya Number Theory 42A38 (Primary) 33C10, 42B05, 26A33, 11P82 (Secondary) We present a systematic analytic study of the $p$-Bessel functions $\mathcal{J}_{ω,φ}^{[p]}$, 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 $\{\mathcal{J}_{ω,φ}^{[p]}\}_{ω\ge0}$ using Erdélyi-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. |
| title | Analytic Study of $p$-Bessel Functions: Fractional Calculus, Integral Representations, and Complex Extensions |
| topic | Number Theory 42A38 (Primary) 33C10, 42B05, 26A33, 11P82 (Secondary) |
| url | https://arxiv.org/abs/2603.21072 |