Exercise 3, § 1

Let K be a field of characteristic 0. One says that an endomorphism U of the algebra K[X,Y] of polynomials in two indeterminates X, Y over K is a composition operator if, for every polynomial $f \in K[X, Y]$, one has, putting $g = U(f)$, that $g(X+S, Y+T) = U_{X,Y}(f(X+S, Y+T))$, where S and T are two other indeterminates.

a) Generalize prop. 1 and th. 1 to these operators. Derive a new proof of the formula $e^{S+T} = e^S e^T$ from this.

For composition operators of the form $D_X^p D_Y^q u(D_X, D_Y)$ where the constant term of the formal series $u$ is not zero, define the Appell polynomials $u_{mn}$ and generalize props. 4, 5 d 6 of VI, p. 273 and 274.

Consider in particular the composition operator $U$ defined by $U(f(X, Y)) = f(X + Y + 1) - f(X, Y + 1) - f(X + 1, Y) + f(X, Y)$; the Appell polynomials corresponding this operator are called Bernoulli polynomials and are denoted by $B_{m,n}$. Show that $u_{m,n}(X, Y) = B_m(X)B_n(Y)$.