The asymptotic expansion of the Humbert hyper-Bessel function

TL;DR

This paper derives the asymptotic expansion of the Humbert hyper-Bessel function for large arguments, focusing on exponentially small contributions, and validates the results with numerical demonstrations.

Contribution

It provides a detailed asymptotic expansion of the Humbert hyper-Bessel function, including exponentially small terms, extending previous methods and applying saddle-point analysis.

Findings

Asymptotic expansion valid as x→+∞

Explicit form of exponentially small contributions

Numerical validation of the asymptotic approximation

Abstract

We consider the asymptotic expansion of the Humbert hyper-Bessel function expressed in terms of a ${}_0F_2$ hypergeometric function by \[J_{m,n}(x)=\frac{(x/3)^{m+n}}{m! n!}\,{}_0F_2(-\!\!\!-;m+1, n+1; -(x/3)^3)\] as $x\to+\infty$, where $m$, $n$ are not necessarily non-negative integers. Particular attention is paid to the determination of the exponentially small contribution. The main approach utilised is that described by the author (J. Comput. Appl. Math. {\bf 234} (2010) 488-504); a leading-order estimate is also obtained by application of the saddle-point method applied to an integral representation containing a Bessel function. Numerical results are presented to demonstrate the accuracy of the resulting compound expansion.