  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.
