  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.
