Lagrange Inversion via Transforms

In [3] we described a technique for solving certain linear operator equations by studying the operator power series de…ned by the system. Essential for obtaining explicit solutions is a Lagrange inversion formula for power series with coe¢ cients in an integral domain K. Such a formula can be found in “Recursive Matrices and Umbral Calculus”by Barnabei, Brini and Nicoletti [1]. J. F. Freeman’s [2] development of a theory of transforms of linear operators on generating functions provides us with a new interpretation of what inversion could mean in general (Theorem 1). We show how special choices of operators and generating functions then produce the desired formula. Lagrange inversion requires imbedding of power series into Laurent series. Therefore, we have to investigate transforms in a slightly more general situation than it was done in [2]. The generalization carefully preserves all the important properties of transforms. These properties are listed in Section 4, omitting most of the straightforward proofs.