The kernel of the radially deformed Fourier transform

The radially deformed Fourier transform, introduced in [S. Ben Said, T. Kobayashi and B. Orsted, Laguerre semigroup and Dunkl operators, Compositio Math.], is an integral transform that depends on a numerical parameter $a \in R^{+}$\mathrm{.}Sofar,onlyfor$a=1$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 $a=2\mathrm{/}n$ with $n\in N$. As a consequence, it is shown that the integral kernel is bounded in dimension 2.