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I'm looking for a reference which discusses the following:

Consider the ODE

$\frac{d^2H}{dz^2} + f(z,x)\frac{dH}{dz} + g(z,x)H(z) = 0$

where

1) $f$ and $g$ depend holomorphically on $x$ and $z$

2) $f/z$ and $g/z^2$ are holomorphic at $\infty$.

As discussed in many books one can find a pair of formal solutions to these which turn out to be asymptotic expansions of actual solutions in appropriate sectors of the complex plane. Presumably, these actual solutions depend holomorphically on $x$ under suitable assumptions, but I cannot find this written down anywhere at this level of generality. Does anyone know a reference or paper which discusses this?

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