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

By the classical formula for the derivative of a composite function, wehave h 1 = f 1 g 1 . Further computations provide the following list of subsequentderivatives of h:(2.31) ⎧h 1 = f 1 g 1 ,⎪⎨h 4 = f 4 (g 1 ) 4 + 6 f 3 (g 1 ) 2 g 2 + 3 f 2 (g 2 ) 2 + 4 f 2 g 1 g 3 + f 1 g 4 ,h 5 = f 5 (g 1 ) 5 + 10 f 4 (g 1 ) 3 g 2 + 15 f 3 (g 1 ) 2 g 3 + 10 f 3 g 1 (g 2 ) 2 +⎪⎩h 2 = f 2 (g 1 ) 2 + f 1 g 2 ,h 3 = f 3 (g 1 ) 3 + 3 f 2 g 1 g 2 + f 1 g 3 ,+ 10 f 2 g 2 g 3 + 5 f 2 g 1 g 4 + f 1 g 5 ,h 6 = f 6 (g 1 ) 6 + 15 f 5 (g 1 ) 4 g 2 + 45 f 4 (g 1 ) 2 (g 2 ) 2 + 15 f 3 (g 2 ) 3 ++ 20 f 4 (g 1 ) 3 g 3 + 60 f 3 g 1 g 2 g 3 + 10 f 2 (g 3 ) 2 + 15 f 3 (g 1 ) 2 g 4 ++ 15 f 2 g 2 g 4 + 6 f 2 g 1 g 5 + f 1 g 6 .Theorem 2.32. For every integer κ 1, the κ-th derivative of the compositefunction h = f ◦ g may be expressed as an explicit polynomial in the partialderivatives of f and of g having integer coefficients:(2.33)d κ hκ∑dx = ∑ ∑ ∑κd=11λ 1