Page 503 - Advanced Linear Algebra

The Umbral Calculus 487

² ³ ~ ² ³ ² ³ b ² ³ ² ³ ~ ² ³

and so ² ³ ~ . Assuming that ² ³ ~ for ~ Á Ã Á c we have

² ³ ~ ² ³ ² ³ b ² ³ ² ³ ~ ² ³

and so ² ³ ~ . Thus, ² ³ ~ Á .

Next, define a linear functional ²!³ by
º ²!³ “ ²%³» ~ Á

Since º ²!³ “ » ~ º ²!³ “ ²%³» ~ and º ²!³ “ ²%³» ~ £ we deduce

that ²!³ is a delta series. Now, the binomial identity gives

&!
º ²!³ “ ²%³» ~ 45 ²&³º ²!³ “ c ²%³»

~

~ 45 ² & ³ c Á

~
~ c ²&³
and so
&! &!

º “ ²!³ ²%³» ~ º “ c ²%³»
&
and since this holds for all , we get ²!³ ²%³ ~ c ²%³ . Thus, ²%³ is the

associated sequence for ²!³ .
)
For part 2 , if ²%³ is a Sheffer sequence, then taking ²!³ ~ &! in Theorem

19.13 gives the Sheffer identity. Conversely, suppose that the Sheffer identity
holds, where ²%³ is the associated sequence for ²!³ . It suffices to show that

²!³ ²%³ ~ ²%³ for some invertible ²!³. Define a linear operator by
;

; ²%³ ~ ²%³

Then
&!
&!
; ²%³ ~ ²%³ ~ ²% b &³

and by the Sheffer identity,

; ²%³ ~ &! ²&³; c ²%³ ~ 45 45 ²&³ c ²%³

~ ~
and the two are equal by part 1 . Hence, commutes with &! and is therefore
)
;
of the form ²!³ , as desired.