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 (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*αk=μ*α⋯*αμ (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 (2n+1)-dimensional Heisenberg group Hn.