Asymptotics for a Variant of the Mittag-Leffler Function

TL;DR

This paper extends the classical Mittag-Leffler function by introducing an exponent to its coefficients, deriving an asymptotic formula in complex sectors, and applying it to factorial sequence properties.

Contribution

It provides a new asymptotic formula for a generalized Mittag-Leffler function, extending previous work and employing Plana's formula with saddle point analysis.

Findings

Derived an asymptotic formula valid in complex sectors.

Extended classical results by Le Roy and Evgrafov.

Re-proved non-holonomicity of powers of factorial sequence.

Abstract

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.