I started with the differential equation \begin{equation} -\sinh^2(x) \left( \frac{d^2}{dx^2} - \left( k - \frac{b}{\tanh(x)} \right)^2\right) \phi_1(x) = \lambda \phi_1(x) \end{equation} and found a solution involving the hypergeometric function \begin{equation} \phi_1(x) = \left(1-\coth(x)\right)^{(k-|b|)/2} \left(1 +\coth(x)\right)^{-(k+|b|)/2}\text{ }_2F_1 \left( \frac{1}{2} - |b| -i \nu, \frac{1}{2} - |b| + i \nu, k-|b|+1, \frac{1-\coth(x)}{2} \right)\,, \qquad 0 < \nu < +\infty \end{equation} where \begin{equation} i\nu = \sqrt{\frac{1}{4}+b^2-\lambda} \end{equation} However, I'm interested in working on a certain limit of this equation, that occurs for small $x$, in which the equation can be reduced to \begin{equation} -x^2 \left( \frac{d^2}{dx^2} - \left( k - \frac{b}{x} \right)^2\right) \phi_2(x) = \lambda \phi_2(x) \end{equation} and we can obtain a solution in terms of the Whittaker function \begin{equation} \phi_2(x) = W_{b, i\nu} (2 |k|x) \end{equation} From this, we would expect to see \begin{equation} \lim_{x \to 0} \phi_1(x) \to \phi_2(x) \end{equation} But I can't seem to find a way to approximate the hypergeometric function into the Whittaker function.
Any idea on how to proceed?