Many physical problems can be effectively solved using series expansions, such as the utilization of Legendre polynomials. Consider the function $\left( 1+\epsilon^2 - 2\epsilon x \right)^\frac{1}{2}$. It can be readily demonstrated that this function can be represented as a series expansion involving Legendre polynomials:
$$ \frac{1}{\left( 1+\epsilon^2 - 2\epsilon x \right)^\frac{1}{2}} = \sum_{n \ge 0} \epsilon^n P_n(x) \, , $$
Here, $P_n(x)$ denotes the Legendre polynomial of order $n$. The expansion proves to be a valuable tool for solving problems related to this function.
Now, let's explore the possibility of finding a similar expansion for the function $f(x,\epsilon) = \left( 1+\epsilon^2 - 2\epsilon x \right)^{-1}$. Although a Taylor expansion can be employed to obtain an initial insight:
$$ f(x,\epsilon) = 1 + 2x\epsilon + \left( 4x^2-1\right)\epsilon^2 + \dots \, , $$ a more general expression does not readily come to mind. Thus, I seek assistance in establishing an analogous expansion in terms of familiar functions for $f(x,\epsilon) = \left( 1+\epsilon^2 - 2\epsilon x \right)^{-1}$. Any help or insights would be highly appreciated.
$$(1-2\epsilon \cos \theta+\epsilon^2)^{-1}=\sum_{n=0}^{\infty}\frac{\sin(n+1)\theta}{\sin \theta}\epsilon^n...$$