From Sequences to Polynomials and Back, via Operator Orderings

Non-commutativity is a common feature in mathematical modeling of reality which, in quantum mechanics, introduces the so-called Heisenberg-Weyl algebra. This new quality does not come without a price − the order of components in operator successions is now relevant and has to be carefully traced in calculations. A traditional solution to this problem is to standardize the notation by fixing the order of operators; that is, to use the normally ordered expansion in powers of the form qp , in which all creation operators stand to the left of the annihilation operators. A word in the letters p’s and q’s is called balanced if it contains the same number of p and q. Theorem 2.5 shows that every balanced word has a representation as a polynomial in z = 12 (qp+pq). C. M. Bender and G. V. Dunne [4] studied operators in symmetrized form (an,k = a ∗ n,n−k, where ∗ denotes complex conjugation),