Exercise 1, § 1

Functions of a Real Variable, VI, 291

Let K be a field of characteristic 0, let U and V two composition operators in K[X], and W = VU = UV; let (u_n), (v_n) (w_n) be the sequences of Appell polynomials corresponding to U, V, W respectively. Show that if p is the order of U then

w_n^{(p)}(X) = \sum_{k=0}^n \binom{n}{k} v_{n-k}(X)V_0(u_k)

w_n(X+Y) = \sum_{k=0}^n \binom{n}{k} u_k(X)v_{n-k}(Y).

Solution

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