Some remarks on the theorems of Wright and Braaksma on the Wright function ${}_p\Psi_q(z)$

TL;DR

This paper numerically investigates the asymptotic behavior of the Wright function, a generalized hypergeometric function, emphasizing the importance of the Stokes phenomenon in understanding its exponentially small terms.

Contribution

It provides a more precise analysis of the Wright function's asymptotics by incorporating the Stokes phenomenon, building on Wright and Braaksma's theorems.

Findings

Enhanced understanding of the Wright function's asymptotics

Demonstrated the significance of the Stokes phenomenon

Provided numerical evidence for theoretical predictions

Abstract

We carry out a numerical investigation of the asymptotic expansion of the so-called Wright function ${}_p\Psi_q(z)$ (a generalised hypergeometric function) in the case when exponentially small terms are present. This situation is covered by two theorems of Wright and Braaksma. We demonstrate that a more precise understanding of the behaviour of ${}_p\Psi_q(z)$ is obtained by taking into account the Stokes phenomenon.