Page 500 - Advanced Linear Algebra

484 Advanced Linear Algebra

)
1 A sequence ²%³ is the associated sequence for ²!³ if and only if

²%³ ~ º ²!³ “ % »%

~ [
)
2 A sequence ²%³ is Sheffer for ² ²!³Á ²!³³ if and only if

c

²%³ ~ º ² ²!³³ ²!³ “ %»%

~ [
)
Proof. We need only prove part 2 . We know that ²%³ is Sheffer for

² ²!³Á ²!³³ if and only if
& ²!³ B ² & ³
~ !
² ²!³³ ~ !
But this is equivalent to
& ²!³ B ² & ³
N % O ~ L “ ! c % M ~ ² & ³
² ²!³³ !
~
Expanding the exponential on the left gives
B º ² ²!³³ ²!³ “ % » B ²&³
c

&~ L ! % M ~ ²&³
c
[ !
~ ~
%
&
Replacing by gives the result.
Sheffer sequences can also be characterized by means of linear operators.
Theorem 19.12 Operator characterization)
²
)
1 A sequence ²%³ is the associated sequence for ²!³ if and only if

)
a ² ³ ~ Á

)

b ²!³ ²%³ ~ c ²%³ for ‚
)
2 A sequence ²%³ is Sheffer for ² ²!³Á ²!³³ for some invertible series ²!³ if

and only if
²!³ ²%³ ~ c ²%³
for all ‚ .
)
Proof. For part 1 , if ²%³ is associated with ²!³ , then

!
² ³ ~ º “ ²%³» ~ º ²!³ “ ²%³» ~ [ Á

and