Asymptotic expansion of the Wright function for large variable and parameter

TL;DR

This paper derives asymptotic expansions for the Wright function when both its variable and parameter are large, using steepest descents, and validates the results with numerical illustrations.

Contribution

It provides new asymptotic formulas for the Wright function for large arguments and parameters, extending previous work with detailed analysis and numerical validation.

Findings

Derived asymptotic expansions for large variable and parameter

Validated expansions with numerical results

Complemented existing literature with new analytical techniques

Abstract

We consider the asymptotic expansion of the Wright function \[W_{\lambda,\mu}(z)=\sum_{n=0}^\infty\frac{z^n}{n! \Gamma(\lambda n+\mu)}\qquad (\lambda>-1)\] for large (positive and negative) variable and large parameter $\mu$. The analysis is based on use of the method of steepest descents applied to a suitable integral representation and, in part, complements the recent work of Ansari and Askari. Numerical results are presented to illustrate the accuracy of the different expansions obtained.