Let $\alpha$ be a real number. We denote $f_{\alpha} : x \longmapsto (1-x)^{-\alpha}$.
State Cauchy's theorem for a scalar first-order linear differential equation and prove that, for all $x \in ]-1,1[$,
$$f_{\alpha}(x) = \sum_{n=0}^{+\infty} L_{n}(\alpha) \frac{x^{n}}{n!}.$$