  For the discrete Laguerre operators we compute explicitly the corresponding
heat kernels by expressing them with the help of Jacobi polynomials. This
enables us to show that the heat semigroup is ultracontractive and to compute
the corresponding norms. On the one hand, this helps us to answer basic
questions (recurrence, stochastic completeness) regarding the associated
Markovian semigroup. On the other hand, we prove the analogs of the
Cwiekel-Lieb-Rosenblum and the Bargmann estimates for perturbations of the
Laguerre operators, as well as the optimal Hardy inequality.
