On certain families of Drinfeld quasi-modular forms

This paper deals with two problems arising in the study of Drinfeld quasi-modular forms. The first problem is to find the maximal order of vanishing at infinity of a non-zero Drinfeld quasi-modular form and leads to the notion of “extremal” quasi-modular form (highest possible order of vanishing for fixed weight and depth). The second problem is determining differential properties of extremal forms, leading to the notion of “differentially extremal” form. From our investigations, we will obtain an upper bound for the order of vanishing at infinity of non-zero Drinfeld quasi-modular forms of small depths. The paper ends with a collection of tools used in the previous parts. The notion of “extremal” form is similar to one introduced by Kaneko and Koike in [M. Kaneko, M. Koike, On extremal quasimodular forms, Kyushu J. Math. 60 (2006) 457–470].