Closed form of the Fourier-Borel kernel in the framework of Clifford analysis

In this paper we derive for the even dimensional case a closed form of the Fourier-Borel kernel in the Clifford analysis setting. This kernel is obtained as the monogenic component in the Fischer decomposition of the exponential function where denotes the standard inner product on the m-dimensional Euclidean space. A first approach based on Clifford analysis techniques leads to a conceptual formula containing the Gamma operator and the so-called Clifford-Bessel function, two fundamental objects in the theory of Clifford analysis. To obtain an explicit expression for the Fourier-Borel kernel in terms of a finite sum of Bessel functions, this formula remains however hard to work with. To that end we have also elaborated a more direct approach based on special functions leading to recurrence formulas for a closed form of the Fourier-Borel kernel