Polynomials and operator orderings

It is shown that there is an exact one-to-one correspondence between all possible sets of polynomials (both orthogonal and nonorthogonal) and rules for operator orderings. Operator orderings that are Hermitian give polynomials with definite parity. Most of the standard classical orthogonal polynomials are associated with operator orderings that are not particularly simple. However, there is a special one-parameter class of Hermitian operator orderings that corresponds to a class of elegant but little-known orthogonal polynomials called continuous Hahn polynomials.