Page 513 - Advanced Linear Algebra

The Umbral Calculus 497

associated sequence for ²!³ is

²%³ ~ Á %

~
then ¢% and so k ¦ ²%³ k ~ is the umbral operator for the
associated sequence

² k ³ % ~ ² % ³ ~ % Á ~ Á ² % ³
~ ~
This sequence, denoted by

² ²%³³ ~ Á ²%³ (19.10 )

~
is called the umbral composition of ²%³ with ²%³ . The umbral operator

~ c ~ ' ² Á % ³ is the umbral operator for the associated sequence %

where
c
%~ ²%³

and so

%~ Á
²%³
~
Let us summarize.

Theorem 19.20
1 The set of umbral operators on is a group under composition, with
)
K
F
k ~ k and c ~

)
2 The set of associated sequences forms a group under umbral composition

² ²%³³ ~ Á ²%³

~
In particular, the umbral composition ² ²%³³ is the associated sequence

for the composition k , that is,

k ¢% ¦ ² ²%³³

The identity is the sequence % and the inverse of ² % ³ is the associated
sequence for the compositional inverse ²!³ .