Generalized Hardy's identity for the astroid-type p-circle lattice point problem

TL;DR

This paper extends Hardy's identity to astroid-type p-circles using generalized Bessel functions, providing new tools for analyzing lattice point problems and error terms in geometric approximations.

Contribution

The paper derives a generalized Hardy's identity for astroid-type p-circles employing generalized Bessel functions and explores related differential formulas involving the Erdélyi-Kober operator.

Findings

Derived a generalized Hardy's identity for p-circles.

Connected differential formulas to the Erdélyi-Kober operator.

Provided a foundation for future error analysis in lattice point problems.

Abstract

Let $r$ be a positive real number and $p$ satisfy $(2/p)\in\mathbb{N}$. Then, we consider the lattice point problem of the closed curves astroid-type $p$-circle $\{x\in\mathbb{R}^{2}|\ |x_{1}|^{p}+|x_{2}|^{p}=r^{p}\}$ which generalize the circle. In investigating the asymptotic behavior of the error term in the area approximation of the circle, G.H. Hardy conjectured an infimum for the evaluation in 1917. One of the grounds for this conjecture is the Hardy's identity, which is a series representation of the term, consisting of the Bessel function of order one and a certain number-theoretic function. In order to investigate an infimum in the error evaluation of the astroid-type $p$-circle, which is unknown in previous studies, in this paper, we derive generalized Hardy's identity for the figures by using generalized Bessel functions. Furthermore, the differential formula for the functions, which is important for the proof of this identity, is closely related to the Erd\'{e}lyi-Kober operator, and this formula and operator are expected to be useful in our future research.