Obtain the power series solution about $t=0$ of

$$\left(1-t^{2}\right) \frac{\mathrm{d}^{2}}{\mathrm{~d} t^{2}} y-2 t \frac{\mathrm{d}}{\mathrm{d} t} y+\lambda y=0$$

and show that regular solutions $y(t)=P_{n}(t)$, which are polynomials of degree $n$, are obtained only if $\lambda=n(n+1), n=0,1,2, \ldots$ Show that the polynomial must be even or odd according to the value of $n$.

Show that

$$\int_{-1}^{1} P_{n}(t) P_{m}(t) \mathrm{d} t=k_{n} \delta_{n m}$$

for some $k_{n}>0$.

Using the identity

$$\left(x \frac{\partial^{2}}{\partial x^{2}} x+\frac{\partial}{\partial t}\left(1-t^{2}\right) \frac{\partial}{\partial t}\right) \frac{1}{\left(1-2 x t+x^{2}\right)^{\frac{1}{2}}}=0,$$

and considering an expansion $\sum_{n} a_{n}(x) P_{n}(t)$ show that

$$\frac{1}{\left(1-2 x t+x^{2}\right)^{\frac{1}{2}}}=\sum_{n=0}^{\infty} x^{n} P_{n}(t), \quad 0<x<1$$

if we assume $P_{n}(1)=1$.

By considering

$$\int_{-1}^{1} \frac{1}{1-2 x t+x^{2}} d t$$

determine the coefficient $k_{n}$.