Laplace Transforms and the Conversion between Ordinary and Exponential Generating Functions
In this brief article, we prove two useful formulas for converting an ordinary generating function (OGF) of a sequence into the corresponding exponential generating function (EGF), and vice versa. These formulas utilize Laplace integral transforms.
Theorem. Let
\(A\left(x\right) = \sum_{k=0}^{+\infty} a_k x^k\) and \(E\left(x\right) = \sum_{k=0}^{+\infty} \frac{a_k}{k!} x^k\)
be the ordinary and exponential generating functions, respectively, of the sequence \(\left[a_n\right]_{n\geq 0}\). Then, the following identities hold:
i) \(E\left(x\right) = \mathcal{L}^{-1} \left[\frac{1}{z}A\left(\frac{1}{z}\right)\right]\left(x\right)\)
ii) \(A\left(x\right) = \frac{1}{x}\mathcal{L} \left[E\left(z\right)\right]\left(\frac{1}{x}\right)\).
Proof.
(i) \(E\left(x\right) = \sum_{k=0}^{+\infty} \frac{a_k}{k!} x^k = \sum_{k=0}^{+\infty} a_k \mathcal{L}^{-1}\left[\frac{1}{z^{k+1}}\right]\left(x\right) = \)
\(\mathcal{L}^{-1}\left[\sum_{k=0}^{+\infty} a_k \frac{1}{z^k \cdot z}\right]\left(x\right) = \mathcal{L}^{-1}\left[\frac{1}{z}A\left(\frac{1}{z}\right)\right]\left(x\right)\)
(ii) \(A\left(x\right) = \sum_{k=0}^{+\infty} a_k x^k = \sum_{k=0}^{+\infty} \frac{a_k}{k!} k! x^k = \frac{1}{x}\sum_{k=0}^{+\infty} \frac{a_k}{k!} k! x^{k+1} = \)
\(\frac{1}{x}\sum_{k=0}^{+\infty} \frac{a_k}{k!} \frac{k!}{\left(\frac{1}{x}\right)^{k+1}} = \frac{1}{x}\sum_{k=0}^{+\infty} \frac{a_k}{k!} \mathcal{L}\left[z^k\right]\left(\frac{1}{x}\right) = \)
\(\frac{1}{x}\mathcal{L}\left[\sum_{k=0}^{+\infty} \frac{a_k}{k!} z^k\right]\left(\frac{1}{x}\right) = \frac{1}{x}\mathcal{L} \left[E\left(z\right)\right]\left(\frac{1}{x}\right)\).
Q.E.D.
References
- Mathematics Stack Exchange, Convert an OGF to an EGF
- Joel L. Schiff, The Laplace Transform: Theory and Applications, Springer-Verlag New York, Inc. (1999).
- Wolfram Research (1999), LaplaceTransform, Wolfram Language function, (updated 2023).
©Stefano Spezia. This work is licensed under a Creative Commons Attribution 4.0 International License
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