Exercise 8, § 4

We denote by $(1 + X)^T$ the element $\exp(T \log (1 + X))$ of $\mathbf{Q}[[X, T]]$. a) Show that $(1 + X)^T = \sum_{n \geq 0} \frac{T(T-1) \ldots (T-n+1)}{n!} X^n$. (The coefficient of $X^n$ is a polynomial in $T$ whose value is known for $T = n \in N.$) b) Show that $(1 + X)^l (1 + X)^T = (1 + X)^{T+l}$. Write down explicitly the identity between binomial coefficients so obtained. Show that $(1 + X + Y + XY)^T = (1 + X)^T (1 + Y)^T$.