Dispersion Estimates for the Discrete Laguerre Operator

We derive an explicit expression for the kernel of the evolution group $\exp (-\mathrm{i}t{H}_0)$ of the discrete Laguerre operator $H_0$ (i.e. the Jacobi operator associated with the Laguerre polynomials) in terms of Jacobi polynomials. Based on this expression we show that the norm of the evolution group acting from ${\mathrm{\ell }}^1$ to ${\mathrm{\ell }}^{\mathrm{\infty }}$ is given by $(1+t^2{)}^{-1\mathrm{/}2}$.