Approximate substitutions and the normal ordering problem

In this paper, we show that the infinite generalised Stirling matrices associated with boson strings with one annihilation operator are projective limits of approximate substitutions, the latter being characterised by a finite set of algebraic equations. 1. Introduction The series of papers [1, 2, 3] had two sequels. First one, algebraic, was the construction of a Hopf algebra of Feynman-Bender diagrams [10, 11] arising from the product formula applied to two exponentials. Second one was the construction and description of one parameter groups of infinite matrices [4, 9] and their link with the combinatorics of so called Sheffer polynomials. The object of this paper is to continue the investigation of those one-parameter groups by highlighting the structure of the group of substitutions. It is shown here how we can see this group as a projective limit of what will be called approximate substitution groups. First, we consider Boson creation and annihilation operators with the commutation relation