The asymptotics of the generalised Bessel function

We demonstrate how the asymptotics for large $\mathrm{\mid }z\mathrm{\mid }$ of the generalised Bessel function $${}_0{\mathrm{\Psi }}_1(z)=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{\mathrm{\Gamma }(an+b)n!},$$ where $a>-1$ and $b$ is any number (real or complex), may be obtained by exploiting the well-established asymptotic theory of the generalised Wright function ${}_p{\mathrm{\Psi }}_q(z)$. A summary of this theory is given and an algorithm for determining the coefficients in the associated exponential expansions is discussed in an appendix. We pay particular attention to the case $a=-1\mathrm{/}2$, where the expansion for $z\to \pm \mathrm{\infty }$ consists of an exponentially small contribution that undergoes a Stokes phenomenon. We also examine the different nature of the asymptotic expansions as a function of $\arg z$ when $-1<a<0$, taking into account the Stokes phenomenon that occurs on the rays $\arg z=0$ and $\arg z=\pm \pi (1+a)$ for the associated function ${}_1{\mathrm{\Psi }}_0(z)$. These regions are more precise than those given by Wright in his 1940 paper. Numerical computations are carried to verify several of the expansions developed in the paper.