Page 515 - Advanced Linear Algebra

The Umbral Calculus 499

Á k Á ² % ³ ~ Á Á % !
~

~ ! Á ² % ³
~
~! ² ²%³³
is Sheffer for ² h ² k ³Á k ³ .
Umbral Shifts and Derivations of the Umbral Algebra

We have seen that an operator on is an umbral operator if and only if its
F
adjoint is an automorphism of . Now suppose that ² ³ is the umbral
B
<
F
shift for the associated sequence ²%³ , associated with the delta series

²!³  <. Then
º d ²!³ “ ²%³» ~ º ²!³ “ ²%³»

~ º ²!³ “ b ²%³»
~² b 1) [ b Á
~ ² c ³[ Á c
~º ²!³ c “ ²%³»

and so
d

²!³ ~ ²!³ c (19.11 )

This implies that

d ´ ²!³ ²!³ µ ~ d ´ ²!³ µ ²!³ b ²!³ d ´ ²!³ µ (19.12 )
and further, by continuity, that

d ´ ²!³ ²!³µ ~ ´ d ²!³µ ²!³ b ²!³´ d ²!³µ (19.13 )
Let us pause for a definition.

Definition Let be an algebra. A linear operator on is a derivation if
C
7
7
C² ³ ~ ²C ³ b Cb
for all Á  7 .

Thus, we have shown that the adjoint of an umbral shift is a derivation of the
)
(
umbral algebra . Moreover, the expansion theorem and 19.11 show that d
<

is surjective. This characterizes umbral shifts. First we need a preliminary result
on surjective derivations.