The asymptotic expansion of a generalised incomplete gamma function

We define a generalised incomplete gamma function Qp(a,z), which coincides with the familiar normalised function Q(a,z)=Γ(a,z)/Γ(a) when the integer parameter p=1. It is shown that the large-z asymptotics of Qp(a,z) in the sector |arg z|<πp consists of p exponential expansions. In Re(z)>0, all these expansions are recessive at infinity and form a sequence of increasingly subdominant exponential contributions.