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We describe the integrand in the martingale (or stochastic integral) representation of a square integrable functional F of a Lévy process in terms of (a derivative or difference operator acting on) a map ßF introduced in Rajeev and Fitzsimmons (Stochastics81, 467–476, 2009). The kernels in the chaos expansion of F are also described in terms of the iterated derivative and difference operators.