A positive product formula of integral kernels of $k$-Hankel transforms

The $k$-Hankel transform $F_{k,1}$ (or the $(k,1)$-generalized Fourier transform) is the Dunkl analogue of the unitary inversion operator in the minimal representation of a conformal group initiated by T. Kobayashi and G. Mano. It is one of the two most significant cases in $(k,a)$-generalized Fourier transforms. We will establish a positive radial product formula for the integral kernels of $F_{k,1}$. Such a product formula is equivalent to a representation of the generalized spherical mean operator in terms of the probability measure ${\sigma }_{x,t}^{k,1}(\xi )$. We will then study the representing measure ${\sigma }_{x,t}^{k,1}(\xi )$ and analyze the support of this measure, and derive a weak Huygens’s principle for the deformed wave equation in $(k,1)$-generalized Fourier analysis.