# Convert an OGF to an EGF

If I have an ordinary generating function (OGF) as a rational polynomial:

$$Q(y) = \frac{f_1(y)}{g_1(y)}$$

Which has an alternative form of

$$Q(y) = \sum_{n=0} a_n y^n$$

How can I compute the exponential generating function (EGF) $$Z(y) = \sum_{n=0} b_n y^n / n!$$ as a rational polynomial? $$Z(y) = \frac{f_2(y)}{g_2(y)}$$

As an explicit solved example, consider the case where $$Q(y) = \frac{yv -1}{y(2+v) -1}$$ The intermediate terms are $$a_n = 2 (v+2) ^ {n-1}$$ Giving as a final answer $$Z(y) = \frac{2 e^{(v+2)y}}{v+2}$$

... ideally, I like to be able to do this by going from one rational polynomial to another without computing $a_n$. We can assume that I can factor the polynomial $g_1(y)$.

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What you want is the Borel transform. This is related to the inverse Laplace transform. Specifically, $Z(s)$ is the inverse Laplace transform of $Q(1/t)/t$.

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This doesn't seem give the right answer - looking it up on Wolfram Alpha gives an extra factor of $v/(v+2)$. It is however close, am I doing something wrong? –  Hooked May 20 '11 at 18:25
Your example wasn't quite right. In the Maclaurin series of $Q(y) = (y v - 1)/(y (v+2) - 1)$, the coefficient of $y^k$ is $a_k = 2 (v+2)^{k-1}$ for $k \ge 1$, but is 1, not $2/(v+2)$, for $k=0$. $Q(1/t)/t = (t-v)/(t(t-v-2)$, whose inverse Laplace transform is $Z(y) = (2 e^{(2+v)y}+v)/(v+2)$. The Maclaurin series for this is indeed $1 + 2 y + \frac{2(v+2)}{2!} y^2 + \frac{2(v+2)^2}{3!} y^3 + \ldots$. –  Robert Israel May 21 '11 at 0:06