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When additive models with more than two covariates are fitted with the backfitting algorithm proposed by Buja et al. [2], the lack of explicit expressions for the estimators makes study of their theoretical properties cumbersome. Recursion provides a convenient way to extend existing theoretical results for bivariate additive models to models of arbitrary dimension. In the case of local polynomial regression smoothers, recursive asymptotic bias and variance expressions for the backfitting estimators are derived. The estimators are shown to achieve the same rate of convergence as those of univariate local polynomial regression. In the case of independence between the covariates, non-recursive bias and variance expressions, as well as the asymptotically optimal values for the bandwidth parameters, are provided.