I'm trying to compute Jacobi fields of the hyperbolic disk $\mathbb{H}^m$ considered as a minimal hypersurface in $\mathbb{H}^{m+1}$ in the half model. References to literature or solutions to the below would be extremely useful:
I take the parameterization of the hemisphere
$$ \mathbb{H}^m = HS^m = \{(x, y_1, \dots, y_{m}, y_{m+1}, \dots, y_n) = (x, \sqrt{1 - x^2} F_{m-1}(\vec{\theta}), y_{m+1} = 0, \dots, y_n = 0) \}$$
where $F_{m-1}(\vec{\theta})$ is the polar parameterization of $S^{m-1}$. In this case, one can compute
$$\Delta_{\mathbb{H}^m} = \left((x \partial_x)^2 - (m-1)(x \partial_x)\right) + \left[\left( (-2x^2) x\partial_x - (x^4) \partial_x^2 \right) \right] + \frac{x^2}{1 - x^2} \Delta_{S^{m-1}}$$ $$J_{\mathbb{H}^{m}} = \Delta_{\mathbb{H}^m} - m$$
We can try to find solutions which are $O(x^{-1})$ as $x \to 0$ by separation of variables. Noting that the eigenvalues of $\Delta_{S^{m-1}}$ are given by $\lambda = k(m+k-2)$, this boils down to solving
$$\left[\left( (x \partial_x)^2 f(x) - (m-1)(x \partial_x) f(x) - m f(x) \right) + \left( -x^4 \partial_x^2 f(x) + x(-2x^2)\partial_x f(x) \right) \right] - \frac{[k(m+k-2)] x^2}{1 - x^2} f(x) = 0$$
Making the subsitution $f(x) = \frac{B_k(x)}{x}$, this simplifies significantly to
$$B_{k,xx} - \frac{m}{x(1-x^2)} B_{k,x} - \frac{k(m+k-2)}{(1 - x^2)^2}B_k = 0$$
I am interested in the case of $k > 0$ - in general, solutions to the above have been difficult to find and most literature references the hypergeometric function. However, when $m = 2$ and $k = 2$, there is an explicit closed form solution of
$$\overline{B}_2(x) = \frac{(1-x)(2x+1)}{1+x}$$
which satisfies $\overline{B}_2(0) = 1$ and is defined on all of $[0,1]$.
My question is: do other closed form solutions exist? I am particular interested in the case when $m$ and $k$ are both even. Ideally the solution would be $P(x) / (1-x^2)^m$ where $P(x)$ is a polynomial, though I don't know how to show this.
