The differential operator with can easily be expanded in terms of the operators by considering its action on
First expand it in terms of the number operator
(For more on this expansion, cf. OEIS-A094638.)
Now consider the generalized Dobinski relation for the Bell / Touchard polynomials defined operationally by with in conjunction with an Euler transformation. Umbral composition of these polynomial operators with an analytic function formally gives
Note that action on of the differential op generates the Newton series interpolation
The same result is achieved by umbral substitution of the falling factorial into the functional series since the falling factorials and the Bell polynomials are an umbral compositional inverse pair, i.e.,
We can derive the well-known formula for the coefficients of the Bell polynomials, i.e., the Stirling numbers of the second kind by choosing then
and we can identify
Applying the general operadic Dobinski relation to the binomial expressions for our op at the top of the page, we obtain
Now using the relation between the Stirling numbers of the first kind and the falling factorials
and the formula for the Stirling numbers of the second kind, we obtain
Written as lower triangular matrices,
with the diagonal matrix
The first few rows of the Stirling matrices are
Examples of the matrix and associated OEIS entries
inverse is padded A049410.
inverse is padded A049404.
inverse is A122848.
inverse is .
inverse is A111596.
inverse is signed, padded A046089.
inverse is signed, padded A049352.
inverse is signed, padded A049353.
Related umbral compositional identities:
The inverse of is simple to derive by inspection. An alternate method incorporating umbral compositional inversion is illustrated below.
Define the signed Lah polynomials through
These polynomials form a self-inverse set under umbral composition; that is,
Then with the falling factorial polynomials
and the rising factorial polynomials
we have, from the Vandermonde-Chu identity, the umbral identity
which, from the sign relations between the factorials, implies
The last relation follows also from the self-inverse property of the Lah polynomials:
Substituting in the last expression restores the original self-inverse formula for the Lah polynomials since the falling factorials and the Bell polynomials are an inverse pair under umbral composition, as mentioned above.
A formula for the Lah polynomials that reflects the formula for comes from substituting into Eqn. I:
The formula for can be derived from the inverse relation between the Bell and falling factorials
and the general relation is
the row polynomials for , which by the generalized Dobinski relation also gives
The umbral inverse of this expression, giving the matrix inverse for , is
This is the umbral compositional inverse of the row polynomials (ordinary generating function), i.e.,
The generalized Dobinski formula gives
Then accumulating the coefficients from the above formulas,
with the matrix rep
with the matrix rep
An equivalent method using the generalized differential shift op
A way to represent umbral substitution is with the generalized shift operators, which, for me, provide some intuition for the simple Euler transformations and the Newton series interpolation as well.
Consider the action of the diff op on the integer power basis:
acting on functions analytic at the origin.
For the diff op we are investigating, is a polynomial in of degree ; therefore, the finite differences higher than that degree vanish, and the last infinite series is truncated to a polynomial in the op basis. That polynomial op may act on functions analytic at other than the origin, such as , which generates a Newton series interpolating the coefficients . Acting on generates the Euler transformation used above.
Joni, Rota, and Sagan in From Sets to Functions: Three Elementary Examples have combinatorial derivations for some of these identities.