Page 501 - Advanced Linear Algebra

The Umbral Calculus 485

º ²!³ “ ²!³ ²%³» ~ º ²!³ b “ ²%³»

~ [ Á b

~ ² c ³[ c Á

~ º ²!³ “ c ²%³»
)
)
)
and since this holds for all ‚ we get 1b . Conversely, if 1a and 1b hold,
then

º ²!³ “ ²%³» ~ º! “ ²!³ ²%³»

~² ³ ² ³
c

~² ³ c Á

~ [ Á
and so ²%³ is the associated sequence for ²!³ .

)
As for part 2 , if ²%³ is Sheffer for ² ²!³Á ²!³³ , then

º ²!³ ²!³ “ ²!³ ²%³» ~ º ²!³ ²!³ b “ ²%³»

~ [ Á b

~ ² c ³[ c Á

~ º ²!³ ²!³ “ c ²%³»
and so ²!³ ²%³ ~ c ²%³ , as desired. Conversely, suppose that

²!³ ²%³ ~ c ²%³
;
and let ²%³ be the associated sequence for ²!³ . Let be the invertible linear

operator on defined by
=
; ²%³ ~ ²%³

Then

; ²!³ ²%³ ~ ; c ²%³ ~ c ²%³ ~ ²!³ ²%³ ~ ²!³; ²%³

and so Lemma 19.5 implies that ; ~ ²!³ for some invertible series ²!³ . Then

º ²!³ ²!³ “ ²%³» ~ º ²!³ “ ²!³ ²%³»

~º! “ ²!³ ²%³»

~² ³ ² ³
c

~² ³ c Á

~ [ Á
and so ²%³ is Sheffer for ² ²!³Á ²!³³ .

We next give a formula for the action of a linear operator ²!³ on a Sheffer
sequence.