Using the Cauchy product of power series, deduce that, for all integers $n$ and all real numbers $\alpha$ and $\beta$, $$L_{n}(\alpha + \beta) = \sum_{k=0}^{n} \binom{n}{k} L_{k}(\alpha) L_{n-k}(\beta).$$
Using the Cauchy product of power series, deduce that, for all integers $n$ and all real numbers $\alpha$ and $\beta$,
$$L_{n}(\alpha + \beta) = \sum_{k=0}^{n} \binom{n}{k} L_{k}(\alpha) L_{n-k}(\beta).$$