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.
Categories: math.CA