On the radially deformed Fourier transform

In this paper we consider the kernel of the radially deformed Fourier transform introduced in the context of Clifford analysis in [10]. By adapting the Laplace transform method from [4], we obtain the Laplace domain expressions of the kernel for the cases of $m=2$ and $m>2$ when $1+c=\frac{1}{n},n\in {\mathbb{N}}_0\mathrm{\backslash }\{1\}$ with $n$ odd. Moreover, we show that the expressions can be simplified using the Poisson kernel and the generating function of the Gegenbauer polynomials. As a consequence, the inverse formulas are used to get the integral expressions of the kernel in terms of Mittag-Leffler functions.