This article is divided in two parts. In the first part we endow a certain ring of "Drinfeld quasi-modular forms" for GL2(Fq[T ]) (where q is a power of a prime) with a system of "divided derivatives" (or hyperderivations). This ring contains Drinfeld modular forms as defined by Gekeler in [6], and the hyperdifferential ring obtained should be considered as a close analogue in positive characteristic of famous Ramanujan's differential system relating to the first derivatives of the classical Eisenstein series of weights 2, 4 and 6. In the second part of this article we prove that, when q = 2, 3, if P is a non-zero hyperdifferential prime ideal, then it contains the Poincaré series h = Pq+1,1 of [6]. This last result is the analogue of a crucial property proved by Nesterenko [11] in characteristic zero in order to establish a multiplicity estimate.Drinfeld quasi-modular forms will have weights, types and depths, the zero depth corresponding to the case of modular forms. A useful result we will then prove is a structure theorem for these quasi-modular forms, similar to the one of Kaneko and Zagier quoted above. Let E be the "false" Eisenstein series as defined in [6, p. 686]. With our definition, this function will be a quasi-modular form of weight 2, depth 1 and type 1. Denote by M ≤l w,m the C-vector space of quasi-modular forms of weight w, type m and depth ≤ l, by M the ring generated by all quasi-modular forms (see § 2 for the precise definitions), and by C [E, g, h] ≤l w,m the subspace of C[E, g, h] generated by the monomials E α g β h γ satisfying the conditions 2α + β(q − 1) + γ(q + 1) = w, α + γ ≡ m (mod q − 1) and α ≤ l.In Section 2 we prove:Theorem 1 The functions E, g, h are algebraically independent over C, and we have M ≤l w,m = C[E, g, h] ≤l w,m and M = C[E, g, h].
