By Ulrich Dempwolff

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So dim V3 = k too. Similar in V4 the coefficients of the Gi ’s can chosen arbitrarily which forces dim V4 = k. For f ∈ V1 we have f (n)X n P (X) = Q(X) n≥0 k f (n)X n + = n≥0 bi f (n)X n+i i=1 n≥0 k = ∗ + ∗X + · · · + ∗X k−1 + f (n + k) + n≥0 39 i=1 bi f (n + k − i) X n+k . e. V1 ⊆ V2 . So even V1 = V2 as dim V1 = dim V2 . Considering V4 we observe r i=1 r i=1 Gi (X)(1 − γi X)−di = Gi (X)Q(X)(1 − γi X)−di Q(X) which shows V4 ⊆ V1 . Again the dimension argument yields V1 = V4 . Elements in V4 are linear combinations of terms of the form X j (1 − γX)−c with 0 ≤ j < c and 0 = γ ∈ C.

76(1969), 871-889. 6 Generating Functions and Recurrences The theme of this section is to turn number sequences into formal power series. Then one manipulates the power series to obtain more information about the number sequence. Definition Let (an )n≥0 be a complex sequence. The ordinary generating function of (an ), abbreviated OGF, is the formal power series ∞ an X n n=0 and the exponential generating function of (an ), abbreviated EGF, is the formal power series ∞ an n X . n! n=0 Examples (a) Set an = 1 for n ≥ 0.

We define a tree Γ0 on [n + 1] by adding n + 1 to the vertex set of Γ and the edges {a1 , n + 1}, . . , {ak , n + 1} to the set of edges. If on the other hand if ∆ is a tree on [n + 1] and N (n + 1) = {a1 , . . , ak } we delete form ∆ the vertex n + 1 and the edges {a1 , n + 1}, . . , {ak , n + 1} and obtain a rooted forest ∆0 on [n] with k connected components. Of course we take the ai ’s as roots and observe that the components are characterized by the ai ’s. Note that we can obtain from two trees isomorphic forests if we neglect the labeling.