Page 505 - Advanced Linear Algebra

The Umbral Calculus 489

which can be rewritten in the more familiar form
B &
&
² b !³ ~ 45 !
~
Of course, this is a formal identity, so there is no need to make any restrictions
on . The binomial identity in this case is
!

²% b &³ ~ 45 ²%³ ²&³ c

~
which can also be written in the form

%b& % &
4 5 4~ 5 4 5
c
~
This is known as the Vandermonde convolution formula .

Example 19.6 The Abel polynomials

(²%Â ³ ~ %²% c ³ c

form the associated sequence for the Abel functional
²!³ ~ !e !

also discussed in Example 19.2. We leave verification of this to the reader. The
generating function for the Abel polynomials is

B &²& c ³ c
& ²!³ ~ !
[
~
Taking the formal derivative of this with respect to gives
&
B ²& c ³²& c ³ c
²!³ & ²!³ ~ !
[
~
which, for &~ , gives a formula for the compositional inverse of the series
!
²!³ ~ ! ,
B ²c ³
c
²!³ ~ !
² c ³[
~

Example 19.7 The famous Hermite polynomials /²%³ form the Appell

sequence for the invertible functional

²!³ ~ !°