REMARKS ABOUT THE CANONICAL FAMILIES OF P OLYNOMIALS THAT GENERATE THE EXPONENTIAL

We study the affine structure of a subset generating all the solutions of the functional equation S(X + Y,T)/S(X, T) is an element of K[[Y, T]] where K is any commutative field and S(X, T) belongs to K[[X, T]] (by combinatorial means, one must assume indeed char(K) = 0 to get non -trivial solutions). The elements of this subset NW' are the formal po wer series S(X,T) = Sigma(n greater than or equal to 0) H-n(X)T-n, whe re the sequence (H-n(X)) is an element of K[X] satisfies: H-0 = 1, H-1 = X and H-n(X(1) +...+ X(k)) = Sigma(alpha 1+...+alpha k=n) H-alpha 1 (X(1))...H-alpha k(X(k)) (n,k greater than or equal to 0). Let define an application gamma in K[X] by gamma(P(X)) = P(X + Y) and let d be th e standard derivation. By using these notations, the classical Taylor' s formula can be written : gamma = exp(Yd). We shall show that there i s a canonical bijection between NW' and the set of endomorphisms f of the K-space K[X] which commute with gamma or d and s.t. f(1) = 0, f(X) = 1. More precisely, for such a f, there exists S(X,T) = Sigma(n grea ter than or equal to 0) H-n(X)T-n is an element of NW' s.t. gamma = Si gma(n greater than or equal to 0) H-n(Y)f(n) (reciprocally, this is a ''Taylor's formula'' which characterizes f or (H-n(X))). The set of th ese endomorphisms f is the orbit under the operation of the group GSF( K) (the group of series B(T) = Sigma(n>0)b(n)T(n) with b(n) is an elem ent of K and b(1) = 1 which are invertible for the composition of the formal power series) on NW' defined by (B, Sigma(n greater than or equ al to 0)H(n)(X)T-n) --> Sigma(n greater than or equal to 0)H(n)(X)(B-1 (T))(n) = Sigma(n greater than or equal to 0)K(n)(X)T-n (and K-n(X) is an element of K[X]). From several independent lemmas on linear algebr a and combinatorial analysis, one get new developments in various doma ins : heights in several variables, geometry of polynomials...