A local central limit theorem on the Laguerre hypergroup

We consider here the Laguerre hypergroup ( K , * α ) , where K = [ 0 , + ∞ [ × R and * α a convolution product on K coming from the product formula satisfied by the Laguerre functions L m ( α ) ( m ∈ N , α ⩾ 0 ). We set on this hypergroup a local central limit theorem which consists to give a weakly estimate of the asymptotic behavior of the convolution powers μ * α k = μ * α ⋯ * α μ (k times), μ being a given probability measure satisfying some regularity conditions on this hypergroup. It is also given a central local limit theorem for some particular radial probability measures on the ( 2 n + 1 ) -dimensional Heisenberg group H n .