Form Sequences to Polynomials and Back, via Operator Orderings

TL;DR

This paper explores the relationship between operator orderings, polynomial representations, and orthogonal polynomials, providing methods to convert between sequences of operators and polynomial forms.

Contribution

It introduces a novel framework connecting operator sequences with polynomial expressions and demonstrates how to derive orthogonal polynomials from these relations.

Findings

Established a method to express operator sequences as polynomials in a new variable.

Derived relations linking polynomial coefficients with operator orderings.

Provided examples of orthogonal polynomials arising from these operator relations.

Abstract

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.