Closed form of the Fourier-Borel kernel in the framework of Clifford analysis (Q692516)

The authors find closed expressions for the monogenic part in the Fischer decomposition of the Fourier kernel, called the Fourier-Bessel kernel. This kernel can be seen to be equal to the sum over the zonal monogenics of all degrees. The first expression is a formal one in terms of a polynomial in the Gamma operator acting on the Clifford-Bessel functions. The second one is obtained through the recursion relations of the Gegenbauer polynomials and yields an expression in terms of a finite sum of Bessel functions of the first kind.