  We consider the gap probability for the Bessel process in the single-time and
multi-time case. We prove that the scalar and matrix Fredholm determinants of
such process can be expressed in terms of determinants of integrable kernels
\`a la Its-Izergin-Korepin-Slavnov and thus related to suitable Riemann-Hilbert
problems. In the single-time case, we construct a Lax pair formalism and we
derive a Painlev\'e III equation related to the Fredholm determinant.
