Page 491 - Advanced Linear Algebra
P. 491
The Umbral Calculus 475
In particular, º % » ~ and so the formal power series representation for
this functional is
B º % » B
²!³ ~ ! ~ ! ~ !
~ ~ ! !
which is the exponential series. If ! is evaluation at , then
! !
~ ² b ³!
and so the product of evaluation at and evaluation at is evaluation at
b .
When we are thinking of a delta series < as a linear functional, we refer to
it as a delta functional . Similarly, an invertible series < is referred to as an
invertible functional. Here are some simple consequences of the development
so far.
Theorem 19.2
)
1 For any < ,
B º ²!³ % »
²!³ ~ !
[
~
)
2 For any F ,
º! ²%³»
²%³ ~ %
[
)
3 For any Á < ,
º ²!³ ²!³ % » ~ 45 º ²!³ % »º ²!³ % c »
~
4) ² ²!³³ deg ²%³ ¬ º ²!³ ²%³» ~
)
5 If ² ³ ~ for all , then
B
L c M ²!³ ²%³ ~ º ²!³ ²%³»
~
where the sum on the right is a finite one.
)
6 If ² ³ ~ for all , then
º ²!³ ²%³»~º ²!³ ²%³» for all ¬ ²%³~ ²%³
)
7 If deg ²%³ ~ for all , then
º ²!³ ²%³»~º ²!³ ²%³» for all ¬ ²!³~ ²!³
