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It is well known that the ordinary and exponential generating functions of a sequence of numbers are related by an integral transform (the Borel transformation).

Does there exist a combinatorial theory of integral transforms? The example above indicates that something might be going on "behind the scenes". Has anyone been able to formulate a precise combinatorial explanation of this phenomenon? If not this one, might other types of integral transforms have combinatorial interpretations?

Thanks for your interest!

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I am not sure that I fully understand the question, but could the following book: G.P. Egorychev, Integral Representation and the Computation of the Combinatorial Sums, AMS, 1984 (Zbl 0453.05001) be relevant? There, the author goes in direction opposite to what you are seemingly interested in: starting from a combinatorial identity, to find its interpretation in terms of integral operators. –  Pasha Zusmanovich Jun 20 '11 at 7:55
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I claim that there is a construction here similar to the one in "Lagrange inversion for species" by Gessel and Labelle, which can explain the general picture.

To every species $S:\mathcal{B}\to \mathcal{B}$ there should correspond a "labelled" version $L(S):\mathcal B\to \mathcal B$ whose exponential generating function is the ordinary generating function of $F$. The construction should start by looking at the span $$\mathcal B \leftarrow \mathcal B\times\mathcal B\rightarrow \mathcal B$$ with maps $p,q$ and introducing a "kernel" on $\mathcal B\times \mathcal B$ in a manner that $L$ can be defined from $$F(\mathcal B)\leftarrow F(\mathcal B\times \mathcal B)\rightarrow F(\mathcal B)$$ by a pull-push formalism.

This kernel $\kappa (X,Y)$ on $F(\mathcal B^2)$ should be a species of 2 sorts, but since labelling is the inverse of $\sum \frac{x^ny^n}{n!}$ it will actually be a virtual species (formal difference of species, so $\kappa$ is a categorification of $e^{-xy}$). Then one has to check that $$L=p _{\ast}(\kappa q^{\ast})$$ or something analogous to it holds. I.e. I suspect that there is a categorification of the Borel (also called Laplace-Carson?) transform which is a natural operation on species, and I believe that the appearances of these integral transforms arise from decategorifications of properties of spans of groupoids.

Unfortunately, I don't know where to find a treatment of this. I was looking a while back for a treatment of the (closely related) Laplace transform along similar lines but couldn't find any references. Since I can't work out details right now, perhaps someone who knows this can elaborate?

Also, there is a section on differential-integral equations with species in "Combinatorial species and tree-like structures" by F. Bergeron, G. Labelle and P. Leroux, which is worth looking at.

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