Page 514 - Advanced Linear Algebra
P. 514
498 Advanced Linear Algebra
)
3 Let K and ! ³ < ² . Then as operators,
c ²!³ ~ ²!³
)
4 Let K and ! ³ < ² . Then
² ²!³³ ~ ²!³
)
Proof. We prove 3 as follows. For any ²!³ < and ²%³ 7 ,
º ²!³ d ²!³ ²%³»~º´ d ²!³µ ²!³ ²%³»
d
~ º d ´ ²!³² c ³ ²!³µ ²%³»
³ ²!³ ²%³»
~ º ²!³² c d
³ ²!³ ²!³ ²%³»
~ º² c d
~ º ²!³ c ²!³ ²%³»
) )
which gives the desired result. Part 4 follows immediately from part 3 since
is composition by .
Sheffer Operators
If ²%³ is Sheffer for ² Á ³ , then the linear operator Á defined by
Á ²% ³ ~ ²%³
is called a Sheffer operator . Sheffer operators are closely related to umbral
operators, since if ²%³ is associated with ²!³ , then
c c
²%³ ~ ²!³ ²%³ ~ ²!³ %
and so
c Á ~ ²!³
It follows that the Sheffer operators form a group with composition
Á k Á ~ c ²!³ c ²!³
c
c
~ ²!³ ² ²!³³
~ ´ ²!³ ² ²!³³µ c k
~ h² k ³Á k
and inverse
c ~ ² ³Á
Á
c
From this, we deduce that the umbral composition of Sheffer sequences is a
Sheffer sequence. In particular, if ²%³ is Sheffer for ² Á ³ and
!²%³ ~ ! ' Á % is Sheffer for ² Á ³, then
