Page 517 - Advanced Linear Algebra

The Umbral Calculus 501

( C and be surjective derivations on .
C
<
Theorem 19.23 The chain rule) Let
Then
C~ ²C ²!³³C

Proof. This follows from

C ²!³ ~ ²!³ c C ²!³ ~ ²C ²!³³C ²!³

and so continuity implies the result.
The chain rule leads to the following umbral result.
Theorem 19.24 If and are umbral shifts, then

~ C k ² ! ³
Proof. Taking adjoints in the chain rule gives
~ k ²C ²!³³ d ~ k C ²!³

We leave it as an exercise to show that C ²!³ ~ ´C ²!³µ c . Now, by taking

²!³ ~ ! in Theorem 19.24 and observing that % ~ % b and so ! ! is
multiplication by , we get
%
Z
~%C ! ~%´C ²!³µ c ~%´ ²!³µ c
!
Applying this to the associated sequence ²%³ for ²!³ gives the following

important recurrence relation for ²%³ .

(
Theorem 19.25 The recurrence formula) Let ²%³ be the associated

sequence for ²!³ . Then
c
Z

1) b ²%³ ~ %´ ²!³µ ²%³

2) ² b % ³ ~ % ´ ² ! ³ µ Z
%
Proof. The first part is proved. As to the second, using Theorem 19.20 we have
c

Z

b ²%³ ~ %´ ²!³µ ²%³

Z
c
~ %´ ²!³µ %
c
Z

~ % ´ ² ²!³³µ %
Z
~ % ´ ²!³µ %
Example 19.9 The recurrence relation can be used to find the associated
!
Z
sequence for the forward difference functional ²!³ ~ c . Since ²!³ ~ ! ,
the recurrence relation is
c!
²%³ ~ % ²%³ ~ % ²% c 1)
b