The asymptotic expansion of the Humbert hyper-Bessel function

We consider the asymptotic expansion of the Humbert hyper-Bessel function expressed in terms of a ${}_0{F}_2$ hypergeometric function by $$J_{m,n}(x)=\frac{(x\mathrm{/}3{)}^{m+n}}{m!n!}{}_0{F}_2(-\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}-;m+1,n+1;-(x\mathrm{/}3{)}^3)$$ as $x\to +\mathrm{\infty }$, 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.