I am reading the article "Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics
in four and higher dimensions", which is on https://arxiv.org/abs/gr-qc/0511111.
I want to derive the formulas (2.16a)-(2.16g) in this article. To do that, we need to solve the continued fractions equation (2.12) by using Wolfram Mathematica, which is
$$
\beta_r-\frac{\alpha_{r-1} \gamma_r}{\beta_{r-1}-} \frac{\alpha_{r-2} \gamma_{r-1}}{\beta_{r-2}-} \ldots \frac{\alpha_0 \gamma_1}{\beta_0}=\frac{\alpha_r \gamma_{r+1}}{\beta_{r+1}-} \frac{\alpha_{r+1} \gamma_{r+2}}{\beta_{r+2}-} \ldots \tag 1
$$ I don't know how to derive the analytic formulas of the series expansion coefficients (2.16a)-(2.16g) from Eq. (1). It is welcome to comment this problem.
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$\begingroup$ As described in doi.org/10.1063/1.523499, putting equation (38) into (37) gives a transcendental equation in which you can identify the coefficients you are seeking, equation (39) and onward. Note that according to doi.org/10.1088/0264-9381/6/7/012, this is only correct up to fifth order. $\endgroup$– MelvinCommented Mar 30 at 19:26
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