Form Sequences to Polynomials and Back, via Operator Orderings

C.M. Bender and G. V. Dunne showed that linear combinations of words \(q^{k}p^{n}q^{n-k}\), where \(p\) and \(q\) are subject to the relation \(qp - pq = \imath\), may be expressed as a polynomial in the symbol \(z = \tfrac{1}{2}(qp+pq)\). Relations between such polynomials and linear combinations of the transformed coefficients are explored. In particular, examples yielding orthogonal polynomials are provided.