I have two functions defined by

$\begin{pmatrix} \mathcal{D}_+(t) \\ \mathcal{D}_-(t) \end{pmatrix} = \begin{pmatrix} W_{\kappa,\mu}(t) \\ W_{-\kappa,\mu}(-t) \end{pmatrix}, $

where $W_{k,\mu}(t)$ is the Whittaker function of the second kind.

From what I have read on the internet, these functions have together an asymptotic expansion around infinity in the open range $-\frac{\pi}{2} < \text{arg}~t < \frac{3\pi}{2}$, as

$\begin{pmatrix} \mathcal{D}_+(t) \\ \mathcal{D}_-(t) \end{pmatrix} \simeq \begin{pmatrix} e^{\frac{-t}{2}}t^{k} \\ e^{\frac{t}{2}}t^{-k}\end{pmatrix}.$

I don't even know if my question is well posed, but I want to compute its monodromy around infinity. I guess that the answer requires the knowledge of the Stokes matrices, but I have no idea how to do. Any suggestion/reference would be highly appreciated!


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