Page 518 - Advanced Linear Algebra

502 Advanced Linear Algebra

Using the fact that ²%³ ~ , we have

²%³ ~ %Á ²%³ ~ %²% c ³Á ²%³ ~ %²% c ³²% c ³
3

and so on, leading easily to the lower factorial polynomials

²%³ ~ %²% c ³Ä²% c b ³ ~ ²%³
Example 19.10 Consider the delta functional

²!³ ~ log ² b !³
!
Since ²!³ ~ c is the forward difference functional, Theorem 19.20 implies
that the associated sequence ²%³ for ²!³ is the inverse, under umbral
composition, of the lower factorial polynomials. Thus, if we write

²%³ ~ :² Á ³%
~
then

%~ :² Á ³²%³
~
The coefficients :² Á ³ in this equation are known as the Stirling numbers of
the second kind and have great combinatorial significance. In fact, :² Á ³ is

the number of partitions of a set of size into blocks. The polynomials ² % ³
are called the exponential polynomials .
The recurrence relation for the exponential polynomials is
b ²%³ ~ %² b !³ ²%³ b ²%³ ~ %² Z ²%³³
Equating coefficients of % on both sides of this gives the well-known formula
for the Stirling numbers
:² b Á ³ ~ :² Á c ³ b :² Á ³
Many other properties of the Stirling numbers can be derived by umbral
means.

Now we have the analog of part 3 of Theorem 19.20.
)
Theorem 19.26 Let be an umbral shift. Then

d ²!³ ~ ²!³ c ²!³