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Suppose that you have an exponential generating function.: $E(z)=\sum_{n=0}^{\infty} \frac{a_{n}z^{n}}{n!}$, and that the definition of $a_{n}$ can be reasonably extended to noninteger arguments. (the Catalan numbers $C_{n}$, would be written in terms of the Gamma function thusly: $C_{n} = \frac{\Gamma(2n+1)}{\Gamma(n+2)\Gamma(n+1)}$, for instance), what then is the combinatorial significance of this integral:

$$U(z)=\int_{0}^{\infty} \frac{a_{v}z^{v}dv}{\Gamma(v+1)}$$ ?

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    You maybe should write $a(v)$; the question is then, how to get a nice form for $a$ if you have $U$?2010-11-11
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    Is there some reason to expect that this integral has combinatorial significance?2010-11-11
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    What is the motivation for such an integral?2011-03-09
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    Do you have one example where you can describe $U$ explicitely?2011-03-10
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    I looked at both the definition of the gamma function $\Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\, dt$ and the Fransén–Robinson constant $F = \int_{0}^\infty \frac{1}{\Gamma(x)}\, dx$, and thought that the way that $e^{-t}$ is in the Gamma function itself sort of resembles the way that the gamma function is used in the constant. This got me thinking about the parallels between summation and integration. And I know that there are many sequences with easy interpolations to the reals. So I wondered what would happen if one, instead of hanging the sequence on a generating function. (cont).2011-03-10
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    What would happen if one were to replace the summation with an integration. From what little I know about analytic combinatorics, I gather that the general term in an integer sequence goes as powers of the roots of the generating function. So would the roots of the "generating integral" say something meaningful combinatorically?2011-03-10

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