A connection between operator orderings and representations of the Lie algebra sl2

We investigate special classes of polynomials in the quantum mechanical position and momentum operators arising from various operator orderings, in particular from the so-called mu-orderings generalizing well-known operator orderings in quantum mechanics such as the Weyl ordering, the normal ordering, etc. Viewing orderings as maps from the polynomial algebra on the phase space to the Weyl algebra generated by the quantum mechanical position and momentum operators we formulate conditions under which these maps intertwine certain naturally defined actions of the Lie algebra sl(2). These conditions arise via certain regularities in coefficients defining the orderings which can nicely be described in terms of some combinatorial objects called here 'inverted Pascal diagrams'. At the end we establish a connection between radial elements in the Weyl algebra and certain polynomials of the 'number operator' expressible in terms of the hypergeometric function. This is related to another representation of sl(2), realized in terms of difference operators.