Following is a recurrence relation written by Robert Israel from Poisson arrivals followed by locking
$u_n(t) = \int_0^{t-nT} \lambda \exp(-\lambda y) u_{n-1}(t-y-T)\, dy$.
The solution he wrote is:
$u_n(t) = 1 - q_n(t-nT) \exp(-\lambda (t-nT))$ where $q_n(s) = \sum_{j=0}^{n-1} (\lambda s)^j/j!$.
I wonder how the recurrence is solved and what kind of method it is?
Thanks in advance!