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 séries procédant suivant les puissances entières et positives d’une variable réelle, 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.