Expression of certain differential operators as rational functions of two fixed operators
Bright, George Walter
Resnikoff, H. L.
Master of Arts
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.