 Laplace transform of a Laguerre polynomial

ID: laplace-transform-of-a-laguerre-polynomial

L_{n} (t) = \sum_{j = 0}^{n} (j n) (- t)^{j} / j!

, apply the Laplace transform to each term, using

L [t^{j}] (p) = j! / p^{j + 1}

for

Re p > 0

L [L_{n}] (p) = \frac{1}{p} \sum_{j = 0}^{n} (j n) (- p^{- 1})^{j} = \frac{1}{p} (1 - p^{- 1})^{n} .

This proves the formula for every nonnegative integer

n

. In particular

L_{2} (t) = 1 - 2 t + t^{2} /2

has transform

(p - 1)^{2} / p^{3}

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