We assume $\alpha = 1$ and use the notation $V_n(z) = U_{n+1}(z,-1)$. Using the expansion of $H_t$ as a power series, deduce that $$\forall n \in \mathbb{N},\, \forall t \in ]0,\pi[, \quad V_n(\cos t) = \frac{\sin((n+1)t)}{\sin t}$$
We assume $\alpha = 1$ and use the notation $V_n(z) = U_{n+1}(z,-1)$. Using the expansion of $H_t$ as a power series, deduce that
$$\forall n \in \mathbb{N},\, \forall t \in ]0,\pi[, \quad V_n(\cos t) = \frac{\sin((n+1)t)}{\sin t}$$