Asymptotics for a variant of the Mittag-Leffler function

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 the work by Le Roy [Valeurs asymptotiques de certaines series procedant suivant les puissances entieres et positives d'une variable reelle, Bull. des Sciences Math. 24, 1900] and Evgrafov [Asimptoticheskie otsenki i tselye funktsii, 3rd ed., Nauka, Moscow, 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.