A symmetric integral identity for Bessel functions with applications to integral geometry

Abstract

In the article of Kunyansky (Inverse Probl 23(1):373–383, 2007) a symmetric integral identity for Bessel functions of the first and second kind was proved in order to obtain an explicit inversion formula for the spherical mean transform where our data is given on the unit sphere in \({\mathbb {R}}^{n}\). The aim of this paper is to prove an analogous symmetric integral identity in case where our data for the spherical mean transform is given on an ellipse E in \({\mathbb {R}}^{2}\). For this, we will use the recent results obtained by Cohl and Volkmer (J Phys A Math Theor 45:355204, 2012) for the expansions into eigenfunctions of Bessel functions of the first and second kind in elliptical coordinates.