3. GENERALITIES ON FEYNMAN GRAPHS 37
3.3. Now that we have defined the functions wγ(P, I) O(U ), we will
arrange them into a formal power series.
Let us define
W (P, I) =
γ
1
|Aut(γ)|
g(γ)wγ(P,
I)
O+(U
)[[ ]]
where the sum is over connected stable graphs γ, and g(γ) is the genus of
the graph γ. The condition that all genus 0 vertices are at least trivalent
implies that this sum converges. Figure 1 illustrates the first few terms of
the graphical expansion of W (P, I).
Our combinatorial conventions are such that, for all a1,...,ak U,

∂a1
· · ·

∂ak
W (P, I) (0) =
γ,φ
g(γ)
|Aut| (γ, φ)
wγ,φ(P, I)(a1,...,ak)
where the sum is over graphs γ with k tails and an isomorphism φ : {1,...,k}

=
T (γ), and the automorphism group Aut(γ, φ) preserves the ordering φ of the
set of tails.
Lemma 3.3.1.
W (0,I) = I.
Proof. Indeed, when P = 0 only graphs with no edges can contribute,
so that
W (0,I) =
i,k
i
|Aut(vi,k)|
wvi,k (0,I).
Here, as before, vi,k is the graph with a single vertex of genus i and valency k,
and with no internal edges. The automorphism group of vi,k is the symmetric
group Sk, and we have seen that wvi,k (0,I) = k!Ii,k. Thus, W (0,I) =

i,k
iIi,k,
which (by definition) is equal to I.
3.4. As before, let P
Sym2
U, and let us write P =

P ⊗P . Define
an order two differential operator ∂P : O(U ) O(U ) by
∂P =
1
2

∂P

∂P
.
A convenient way to summarize the Feynman graph expansion W (P, I) is
the following.
Lemma 3.4.1.
W (P, I) (a) = log {exp( ∂P ) exp(I/ )} (a)
O+(U
)[[ ]].
The expression exp( ∂P ) exp(I/ ) is the exponential of a differential
operator on U applied to a function on U; thus, it is a function on U.
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