  We consider the circular unitary ensemble with a Fisher-Hartwig singularity
of both jump type and root type at $z=1$. A rescaling of the ensemble at the
Fisher-Hartwig singularity leads to the confluent hypergeometric kernel. By
studying the asymptotics of the Toeplitz determinants, we show that the
probability of there being no eigenvalues in a symmetric arc about the
singularity on the unit circle for a random matrix in the ensemble can be
explicitly evaluated via an integral of the Hamiltonian of the coupled
Painlev\'{e} V system in dimension four. This leads to a Painlev\'e-type
representation of the confluent hypergeometric-kernel determinant. Moreover,
the large gap asymptotics, including the constant terms, are derived by
evaluating the total integral of the Hamiltonian. In particular, we reproduce
the large gap asymptotics of the confluent hypergeometric-kernel determinant
obtained by Deift, Krasovsky and Vasilevska, and the sine-kernel determinant as
a special case, including the constant term conjectured earlier by Dyson.
