We assume $\alpha = 1$ and use the notation $V_n(z) = U_{n+1}(z,-1)$. Let $t \in ]0,\pi[$. Show that the function $$H_t : x \mapsto \frac{1}{1 - 2x\cos(t) + x^2}$$ is expandable as a power series on $]-1,1[$.
We assume $\alpha = 1$ and use the notation $V_n(z) = U_{n+1}(z,-1)$. Let $t \in ]0,\pi[$. Show that the function
$$H_t : x \mapsto \frac{1}{1 - 2x\cos(t) + x^2}$$
is expandable as a power series on $]-1,1[$.