Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

348we observe that the following induction relations hold:(4.23)h 2 = F 2 (h 1 ) ,h 3 = F 3 (h 2 ) ,. . . . . . . . . . . . . . .h λ = F λ (h λ−1 ) .To obtain the explicit version of the Faà di Bruno in the case of one variable xand several variables (y 1 , . . ., y m ), it suffices to extract from the expressionof Y j κ provided by Theorem 4.18 only the terms corresponding to µ 1λ 1 +· · · + µ d λ d = κ, dropping all the X terms. After some simplifications andafter a translation by means of an elementary dictionary, we may formulatea statement.Theorem 4.24. For every integer κ 1, the κ-th partial derivative of thecomposite function h = h(x) = f (g 1 (x), . . .,g m (x)) with respect to x maybe expressed as an explicit polynomial depending on the partial derivativesof f, on the derivatives of g and having integer coefficients:(4.25)d κ hκ∑dx = κd=1∑1λ 1