Polynomials on $x\in[-1,1]$ can be written as an expansion in the Legendre polynomials $P_l(x)$. Is it possible to expand more general types of function on this interval in terms of these polynomials. In particular would it be possible to write a function such as $f(x) = \frac{1}{x-1}$ which is divergent at the boundary of the interval, as a series in Legendre polynomials?
My own answer to this would be that such an expansion is not possible for the $f(x)$ given above as the integral to find the coefficient of $P_0 (x)$ in such an expansion for example does not converge or even have a principal value. But I want to confirm I'm not missing any work-around or if there's a way to express this function on the subset $x\in [-1,1)$ as a series in Legendre polynomials.