The kernel of the radially deformed Fourier transform

The radially deformed Fourier transform, introduced in Ben Said et al. [Laguerre semigroup and Dunkl operators. Compos. Math. 2012;148:1265-1336], is an integral transform that depends on a numerical parameter a(+). So far, only for a=1 and a=2, the kernel of this integral transform is determined explicitly. In the present paper, explicit formulas for the kernel of this transform are obtained when the dimension is even and a=2/n with nN. As a consequence, it is shown that the integral kernel is bounded in dimension 2