  For $V(z)$, analytic in a neighborhood of $0\in\mathbb{C}$, $V(0) = 0$,
$V'(0)\ne0$, there is an associated sequence of polynomials, \textsl{canonical
polynomials}, that is a generalized Appell sequence with lowering operator
$V(d/dx)$. Correspondingly, the inverse function $Z(v)$ has an associated
canonical polynomial sequence. The coefficients of the two sequences form two
mutually inverse infinite matrices. We detail the operator calculus for the
pair of systems, illustrating the duality of operators and variables between
them. Examples are presented including a connection between Gegenbauer and
Bessel polynomials. Touchard polynomials appear as well. Alternative settings
would include an umbral calculus approach as well as the Riordan group. Here we
use the approach through operator calculus.
