Asymptotics of the Wright function ${}_1\Psi_1(z)$ on the Stokes lines

We investigate a particular aspect of the asymptotic expansion of the Wright function ${}_1\Psi_1(z)$ for large $|z|$. The form of the exponentially small expansion associated with this function on certain rays in the $z$-plane (known as Stokes lines) is discussed. The main thrust of the paper is concerned with the expansion in the particular case when the Stokes line coincides with the negative real axis $\arg\,z=\pi$. Some numerical examples which confirm the accuracy of the expansion are given.