  We generalize the Mittag-Leffler function by attaching an exponent to its
Taylor coefficients. The main result is an asymptotic formula valid in sectors
of the complex plane, which extends work by Le Roy [Bull. des sciences math.
24, 1900] and Evgrafov [Asimptoticheskie otsenki i tselye funktsii, 1979]. It
is established by Plana's summation formula in conjunction with the saddle
point method. As an application, we (re-)prove a non-holonomicity result about
powers of the factorial sequence.
