Graduate Recurrence relations for series solution of differential equation

[dim_d00m]

TL;DR

I'm looking for a three-term relation between the coefficients of a series solution to the spheroidal harmonics defining equation.

I am currently looking at section IIA of the following paper: https://arxiv.org/pdf/gr-qc/0511111.pdf. Eq. (2.5) proposes an ansatz to solve the spheroidal wave equation (2.1). This equation is
$$ \dfrac{d}{dx} \left((1-x^2) \dfrac{d}{dx}S_{lm} \right) + \left(c^2x^2 + A_{lm} - \dfrac{m^2}{1-x^2}\right)S_{lm}=0,$$

and the ansatz is

$$S_{lm}(x)= e^{cx}(1-x^2)^{|m|/2} \sum_{p=0}^{+\infty} a_p (1+x)^p.$$

Upon differentiating this ansatz and some algebra, I find the following equation:

$$ \sum_{p=0}^{+\infty} a_p (1+x)^p \left[ -2c(1+|m|)x(1+x^2) + c^2(1-x^2)^2 + |m|\left( (|m|+1)x^2 -1 \right) +(1-x^2)(c^2x^2+A_{lm}) -m^2 \right] + \sum_{p=0}^{+\infty} (p+1) a_{p+1} (1+x)^p \left[2x(1-x^2)^2 - 2(1+|m|)(1-x^2) \right] +\sum_{p=0}^{+\infty} (p+2)(p+1) a_{p+2} (1+x)^p (1-x^2)^2 =0. $$

Expanding and grouping powers of $x$, I get
$$\sum_{p=0}^{+ \infty} (1+x)^p \left[a_p[A_{lm} + c^2 - m(m+1) -2xc(m+1) + x^2(m(m+1)-c^2 -A_{lm}) + 2x^3 c (m+1)] \right]\\
+ 2(p+1) a_{p+1} [c- (m+1)x - 2cx^2 + (m+1)x^3 + cx^4] + (p+2)(p+1)a_{p+2}[1-2x^2+x^4]=0.$$

I have no idea how to proceed further. I've tried to look at equating the coefficients of the various powers of $x$ to 0, but I don't get anything like the recurrence relation outlined in Eqs. (2.6) and (2.7) of said paper. Any help would be much appreciated!

 

[Paul Colby]

Welcome to PF!

The basic idea with series solutions is to substitute into the DE, collect like powers of $x$ , and then equate the coefficient of each power of $x$ to zero. Doing this leads to the mess you've shown. Now, the authors have reduced this with perseverance (and or a computer algebra package) leading to the recurrence relations (2.6) and (2.7). Okay, starting with these recurrence relations, one needs to specify $a_0$ then use (2.6) to get $a_1$. From here (2.7) may be used to find $a_2$ and so on. Sadly, this is where the real work starts.

The differential equation (2.1) has singular points at $x = \pm 1$. The expansion, (2.5) is being taken around $x=-1$ so you will get an analytic solution valid in a region around $-1$. The problem is the point $x=1$. For almost all values of $A_{lm}$ the series will diverge at $x=1$. The series will only converge for eigenvalues. Now, they solve this problem in equation (2.9). I have no clue how they arrived at this continued fraction. I'd be interested to find out.