  We study the finite-temperature deformation of the discrete Bessel point
process. We show that its largest particle distribution satisfies a reduction
of the 2D Toda equation, as well as a discrete version of the
integro-differential Painlev\'e II equation of Amir-Corwin-Quastel, and we
compute initial conditions for the Poissonization parameter equal to 0. As
proved by Betea and Bouttier, in a suitable continuum limit the last particle
distribution converges to that of the finite-temperature Airy point process. We
show that the reduction of the 2D Toda equation reduces to the Korteweg-de
Vries equation, as well as the discrete integro-differential Painlev\'e II
equation reduces to its continuous version. Our approach is based on the
discrete analogue of Its-Izergin-Korepin-Slavnov theory of integrable operators
developed by Borodin and Deift.
