We construct explicit representations of the Heisenberg-Weyl algebra [P,M]=1 in terms of ladder operators acting in the space of Sheffer-type polynomials. Thus we establish a link between the monomiality principle and the umbral calculus. We use certain operator identities which allow one to evaluate explicitly special boson matrix elements between the coherent states. This yields a general demonstration of boson normal ordering of operator functions linear in either creation or annihilation operators. We indicate possible applications of these methods in other fields.

9 pages