Page 497 - Advanced Linear Algebra

The Umbral Calculus 481

º ²!³ ²!³ “ ²%³» ~ [ Á (19.5)

for all Á ‚ .
Proof. The uniqueness follows from Theorem 19.2. For the existence, if we set

²%³ ~ % Á

~
and
B

²!³ ²!³ ~ !
Á
~
(
)

where Á £ , then 19.5 is
B
~ L ! c % M
[ Á Á Á
~ ~
B
~ º ! c % Á Á »
~ ~

~ Á Á [
~
Taking ~ we get

~ Á
Á
For ~ c we have

~ c Á c Á c ² c ³[ b c Á Á [
. By
and using the fact that Á ~ ° Á we can solve this for Á c
successively taking ~ Á c Á c Á Ã we can solve the resulting
of the sequence ² % . ³
equations for the coefficients Á
)
(
Definition The sequence ²%³ in 19.5 is called the Sheffer sequence for the

ordered pair ² ²!³Á ²!³³ . We shorten this by saying that ²%³ is Sheffer for

² ²!³Á ²!³³.
Two special types of Sheffer sequences deserve explicit mention.
Definition The Sheffer sequence for a pair of the form ² Á ²!³³ is called the
associated sequence for ²!³ . The Sheffer sequence for a pair of the form
² ²!³Á !³ is called the Appell sequence for ²!³.