Solving for the recursion relation of a second order ODE

Question

I want to solve for the asymptotic solution of the following differential equation

$$ \left(y^2+1\right) R''(y)+y\left(2-q \left(b \sqrt{y^2+1}\right)^{-q}\right) R'(y)-l (l+1) R(y)=0$$

as $y\rightarrow \infty$, where $q>0$. I did the standard way by obtaining a series solution by the Frobenius method prescription in the form

$$R(y)=\sum_{n=0}^\infty \frac{a_{n}}{y^{n+k}}$$ where $k=l+1$ is the indicial exponent. I had difficulty finding for a recurrence relation for the coefficients $a_n$ for arbitrary value of the parameter $q$. Right now, I am just doing the brute force method of solving individual $a_n$ for every value of $q$. But I am just wondering whether the recurrence relation is possible to solve. Can this be solved analytically?

Answer 1

First, we take the most complicated term and expand it in a series around $y \to \infty$:

$$\left(\sqrt{y^2+1}\right)^{-q}= \frac{1}{y^q} {_2 F_1} \left(\frac{q}{2},1;1;-\frac{1}{y^2} \right)= \sum_{k=0}^\infty \frac{(-1)^k (q/2)_k}{k!} \frac{1}{y^{2k+q}}$$

Where $(p)_k=p(p+1)(p+2) \cdots (p+k-1)$ is the Pochhammer symbol, and I'm using hypergeometric series instead of binomial series because it's more convenient.

Then we divide the equation by $y^2$ and obtain:

$$\left(1+\frac{1}{y^2}\right) R''(y)+\left(\frac{2}{y}-\frac{q}{b^q} \sum_{k=0}^\infty \frac{(-1)^k (q/2)_k}{k!} \frac{1}{y^{2k+q+1}} \right) R'(y)-\frac{l (l+1)}{y^2} R(y)=0$$

Now substitute:

$$R(y)=\sum_{n=0}^\infty \frac{a_{n}}{y^{n+\alpha}} \\ R'(y)=-\sum_{n=0}^\infty \frac{(n+\alpha) a_{n}}{y^{n+\alpha+1}} \\ R''(y)=\sum_{n=0}^\infty \frac{(n+\alpha)(n+\alpha+1) a_{n}}{y^{n+\alpha+2}}$$

We obtain:

$$\sum_{n=0}^\infty \frac{(n+\alpha)(n+\alpha+1) a_{n}}{y^{n+\alpha+2}}+\sum_{n=0}^\infty \frac{(n+\alpha)(n+\alpha+1) a_{n}}{y^{n+\alpha+4}}-2\sum_{n=0}^\infty \frac{(n+\alpha) a_{n}}{y^{n+\alpha+2}}+ \\ + \frac{q}{b^q} \sum_{n=0}^\infty \sum_{k=0}^\infty \frac{(-1)^k (q/2)_k}{k!} \frac{(n+\alpha) a_{n}}{y^{2k+n+q+\alpha+2}}-l (l+1) \sum_{n=0}^\infty \frac{a_{n}}{y^{n+\alpha+2}}=0$$

Where $\alpha$ is an unknown exponent.

Multiplying by $y^{\alpha+2}$, we have:

$$\sum_{n=0}^\infty \frac{(n+\alpha)(n+\alpha+1) a_{n}}{y^{n}}+\sum_{n=0}^\infty \frac{(n+\alpha)(n+\alpha+1) a_{n}}{y^{n+2}}-2\sum_{n=0}^\infty \frac{(n+\alpha) a_{n}}{y^{n}}+ \\ + \frac{q}{b^q} \sum_{n=0}^\infty \sum_{k=0}^\infty \frac{(-1)^k (q/2)_k}{k!} \frac{(n+\alpha) a_{n}}{y^{2k+n+q}}-l (l+1) \sum_{n=0}^\infty \frac{a_{n}}{y^{n}}=0$$

Let's assume $q$ is a positive integer, so $q=1,2,3,\ldots$. Then we can collect the powers of $y$.

$$\sum_{n=0}^\infty \frac{(n+\alpha)(n+\alpha+1) a_{n}}{y^n}+\sum_{n=2}^\infty \frac{(n+\alpha-2)(n+\alpha-1) a_{n-2}}{y^n}-2\sum_{n=0}^\infty \frac{(n+\alpha) a_{n}}{y^n}+ \\ + \frac{q}{b^q} \sum_{k=0}^\infty \sum_{n=2k+q}^\infty \frac{(-1)^k (q/2)_k}{k!} \frac{(n-2k-q+\alpha) a_{n-2k-q}}{y^n}-l (l+1) \sum_{n=0}^\infty \frac{a_{n}}{y^n}=0$$

Since $q>0$, it's clear that the indical equation reads:

$$\alpha(\alpha+1)-2\alpha-l(l+1)=0$$

$$\alpha(\alpha-1)=l(l+1)$$

So we have:

$$\alpha_{1,2}= \frac{1}{2} \pm \sqrt{l(l+1)+\frac{1}{4}}$$

The recurrence relation in general should probably look like this (check for a few values of $q$ and $n$ though, because I may have made a mistake in collecting the terms):

$$ \left((n+\alpha-1)(n+\alpha)-\alpha(\alpha-1) \right) a_{n}+(n+\alpha-2)(n+\alpha-1) a_{n-2} + \\ + \frac{q}{b^q} \sum_{k=0}^{\lfloor \frac{n+q}{2} \rfloor} \frac{(-1)^k (q/2)_k}{k!} (n-2k-q+\alpha) a_{n-2k-q} =0$$

Where we obviously take $a_{-1}=a_{-2}=a_{-3}= \dots=0$.

I have replaced $l(l+1)$ by $\alpha(\alpha-1)$, which makes the expression work for the both roots of the indical equation.

For $q$ not an integer, I'm not sure what to do.