I'm working on an eigenvalue problem involving 2 operators, which we will call $\hat{A}$ and $\hat{B}$. They are related for very small values on a coordinate $x \approx 0$, in which $\hat{A}$ transforms into $\hat{B}$, because the former has hyperbolic sine terms that reduce into $\sinh(x) \approx x$ in this limit, resulting in $\hat{B}$. As such we expect for the eigenfunctions of $\hat{A}$ to also transform accordingly into the eigenfunctions of $\hat{B}$, for very small values of $x$.
The eigenfunctions for $\hat{A}$ and $\hat{B}$, respectively are $f(x)$ and $g(x)$ given by: \begin{equation} f(x) = C_1 e^{x(\alpha + \beta)} (1-e^{2x})^{\frac{1}{2} + i \nu} \text{ }_2F_1\left(\alpha + \frac{1}{2} + i \nu,\beta + \frac{1}{2} + i \nu; 1 + \alpha + \beta;e^{2x}\right) \end{equation}
\begin{equation} g(x) = C_2 W_{\beta, i\nu} (2|\alpha| x) \end{equation}
However, I'm having trouble demonstrating that for $x \approx 0$, \begin{equation} f(x) = g(x) \end{equation} In particular, I've been trying to use the integral representations of the Hypergeometric function and applying the limit under consideration but so far I haven't been able to transform it into a Whittaker representation.