Recurrence relations for series solution of differential equation

In summary, the paper proposes an ansatz to solve the spheroidal wave equation. This equation has singular points at ##x = \pm 1##. The series solutions for this equation only converge around eigenvalues. The authors solve this problem by continued fractionation.
  • #1
dim_d00m
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TL;DR Summary
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!
 
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  • #2
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.
 

What is a recurrence relation for series solution of a differential equation?

A recurrence relation is a mathematical equation that defines a sequence of terms based on previous terms. In the context of series solution of a differential equation, a recurrence relation is used to find the coefficients of a power series solution by relating each term to the previous term.

Why is a recurrence relation used for series solution of a differential equation?

A recurrence relation is used because it allows for an infinite number of terms to be calculated without having to explicitly write out each term. This is especially useful for solving differential equations, which often have an infinite number of solutions.

How do you determine the recurrence relation for a given differential equation?

The recurrence relation is determined by substituting the power series solution into the differential equation and equating coefficients of like powers of the independent variable. This process results in a recursive formula that relates each term to the previous term.

What is the order of a recurrence relation for series solution of a differential equation?

The order of a recurrence relation is determined by the highest power of the independent variable in the equation. For example, if the recurrence relation is of the form an+1 = 2an, then it is a first-order recurrence relation.

Can a recurrence relation be used to find an exact solution to a differential equation?

No, a recurrence relation can only be used to find an approximate solution to a differential equation. The power series solution obtained from the recurrence relation will only be an approximation of the exact solution, which can be found using other methods such as separation of variables or variation of parameters.

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