I am not sure if I should ask this question here or on the physics site; pleaase move it if it does not fit the site guide-lines.
I am reading this paper, in partitcular I am reading the Method of continued fraction at pag. 13. I find it confusing and I do not understand the notation, what do the indices on $c_{i,j}^{(N)}(\omega)$ mean? How does it pass from $c^{(N)}$ to $c^{(k)}$ to $c^{(3)}$? What is the Gauss elimination for recurrence relation?
Can anybody help?
EDIT as requested I try to summarize here the content of the paragraph.
Consider a differential equation of the form: $$\left(\dfrac{d^2}{dr^2}+p(r)\dfrac{d}{dt}+q(r)\right)R(r)=0$$ consider the case in which it has two singular points $r_+$ and $r_\infty$ which could be finite (and we have a cosmological horizon) or infinite. A solution has the form: $$R(r)=\left( \dfrac{r-r_\infty}{r-r_0} \right)^{i \Omega}\left( \dfrac{r-r_+}{r-r_0} \right)^{-i a} \sum_{k=0}^\infty b_k \left( \dfrac{r-r_+}{r-r_0}\dfrac{r_\infty -r_0}{r_\infty-r_+} \right)^k$$ $$R(r)=e^{i\Omega r}(r-r_0)^\sigma \left( \dfrac{r-r_+}{r-r_0} \right)^{-i a}\sum_{k=0}^\infty b_k\left( \dfrac{r-r_+}{r-r_0} \right)^{k}$$ in the two cases ($\Omega$, $\sigma$ and $a$ are defined in order to satisfy the dif. equation and $r_0$ is a parameter). Substituting $R$ into the diff equation. we get a $N$-term recurrence relation: $$\sum_{j=0}^{min(N-1,i)}c_{j,i}^{(N)}(\omega)b_{i-j}=0 \qquad 1)$$ for $i>0$. As the paper says we decrease the recurrence relation by one, i.e. we find $c^{(k)}_{j,i}$ which satisfies the relation $$\sum_{j=0}^{min(k-1,i)}c_{j,i}^{(k)}(\omega)b_{i-j}=0$$ we can rewrite the above expression as $$\dfrac{c_{k,i}^{(k)}(\omega)}{c^{(k)}_{j-1,i-1}(\omega)}\sum_{j=1}^k c_{j-1,i-1}^{(k)}(\omega)b_{i-j}=0 \quad 2)$$ Subtracting 2) by 1) we get the relations: $$c_{j,i}^{(k)}(\omega)=c_{j,i}^{(k)}(\omega)\quad for \; j=0 \;or\; i<k$$ $$c_{j,i}^{(k)}(\omega)=c_{j,i}^{(k+1)}(\omega)-\dfrac{c_{k,i}^{(k+1)}(\omega)c_{j-1,i-1}^{(k)}(\omega)}{c_{k-1,i-1}^{(k)}(\omega)}$$ This procedure is called Gaussian elimination and allows to determine the coefficients in the three term recurrence relation
$$c_{0,1}^{(3)}b_i+c_{1,j}^{(3)}b_{i-1}+c_{2,i}^{(3)}b_{i-2}=0 \quad i>1$$ $$c_{0,1}^{(3)}b_1+c_{1,1}^{(3)}b_0=0$$
As I said the notation confuses me and I do not understand what the Gaussian elimination consists in.