The Chebisheff DE is...
$\displaystyle y^{\ ''} - \frac{x}{1-x^{2}}\ y^{\ '} + \frac{\alpha^{2}}{1-x^{2}}\ y= y^{\ ''} + p(x)\ y^{\ '} + q(x)\ y=0$ (1)
If $x_{0}$ is a singularity of p(x) and q(x) and both the limits...
$\displaystyle \lim_{x \rightarrow x_{0}} (x-x_{0})\ p(x)$
$\displaystyle \lim_{x \rightarrow x_{0}} (x-x_{0})^{2}\ q(x)$ (2)
... exist finite, then $x_{0}$ is a regular singular point. You can verify that $x_{0}=1$ and $x_{0}=-1$ are both regular singular points...
Kind regards
$\chi$ $\sigma$