Expression of certain differential operators as rational functions of two fixed operators

[摘要] It is known that the set of differential operators which map T-automorphic forms of weight k into T-automorphic forms of weight k, for all subgroups TcG = the group of fractional linear transformations of the upper half-plane, has countably many generators, say Dn, n=2,3,4,..., where the highest order derivative of f appearing in the expression Dnf is n. We will show that the first two operators, D2 and D3, are enough to generate each of the Dn as a rational expression of compositions of these two fixed operators. In addition, an algorithm will be described which will calculate a specific rational expression equivalent to Dnf, for all n. The denominator, 6n, of this expression is a polynomial in f and (Do)f, where denotes operator composition, for all positive integers s such that 2sn. Then 6nDnf is a polynomial in f and (D3x)r x (D2x)sf, where r is 0 or 1, and s is such that 3r+2sn. Let P be this polynomial. Then the coefficients of P are calculated by applying (6n)(Dnf) = P to certain suitably chosen functions f and using Crairter;;s rule. The impracticality of the algorithm is demonstrated for the case n=4.

[发布机构] Rice University