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

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 Erdélyi-Kober operator, and this formula and operator are expected to be useful in our future research.