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

316(y 1 ) 5 are simple. Indeed, by inspecting the first terms in the expressions ofY 1 , Y 2 , Y 3 , Y 4 and Y 5 , we of course recognize the binomial coefficients.In general:Lemma 2.19. For κ 1,(2.20) ⎧κ∑[( ( ) ]⎪⎨κ κY κ = Y x κ + Yλ)x κ−λ y − X λλ − 1 x κ−λ+1 y (y λ−1 1 ) λ +λ=1⎪⎩+ [−X y κ] (y 1 ) κ + remainder,where the term remainder collects all remaining monomials in the pure jetvariables.In addition, let us remind what we have observed and used in a previousco-signed work.Lemma 2.21. ([GM2003a], p. 536) For κ 4, nine among the monomialsof Y κ are of the following general form:(2.22) ⎧Y κ = Y x κ + [ Cκ 1 Y x κ−1 y − X x κ]y1 + [ Cκ 2 Y x κ−2 y − Cκ 1 X ]x κ−1 y2 +⎪⎨ + [ Cκ 2 Y x 2 y − Cκ 3 X x 3]yκ−2 + [ Cκ 1 Y xy − Cκ 2 X x 2]yκ−1 +⎪⎩+ [ C 1 κ Y y 2 − κ 2 X xy]y1 y κ−1 + [ −C 2 κ X y]y2 y κ−1 ++ [ Y y − C 1 κ X x]+[−C1κ+1 X y]y1 y κ + remainder,where the term remainder denotes all the remaining monomials, and whereCκ λ := κ!is a notation for the binomial coefficient which occupies less(κ−λ)! λ!space in Latex “equation mode” than the classical notation( κ(2.23).λ)Now, we state directly the final theorem, without further inductive or intuitiveinformation.Theorem 2.24. For κ 1, we have:(2.25)∑κ+1∑Y κ = Y x κ +d=11λ 1