Page 494 - Advanced Linear Algebra

478 Advanced Linear Algebra

by defining the linear operator ²!³¢ F ¦ F by
B
³

²
²!³ ²%³ ~ ´! ²%³µ ~ ²%³
! !
~ ‚
the latter sum being a finite one. Note in particular that

²!³% ~ 45 % c (19.3 )

~
With this definition, we see that each formal power series  < plays three
roles in the umbral calculus, namely, as a formal power series, as a linear
functional and as a linear operator. The two notations º ²!³ “ ²%³» and
²!³ ²%³ will make it clear whether we are thinking of as a functional or as an

operator.
<
It is important to note that ~ in if and only if ~ as linear functionals,
which holds if and only if ~ as linear operators. It is also worth noting that
´ ²!³ ²!³µ ²%³ ~ ²!³´ ²!³ ²%³µ
and so we may write ²!³ ²!³ ²%³ without ambiguity. In addition,
²!³ ²!³ ²%³ ~ ²!³ ²!³ ²%³

for all Á  < and  F .

When we are thinking of a delta series as an operator, we call it a delta

operator. The following theorem describes the key relationship between linear
functionals and linear operators of the form ²!³ .

Theorem 19.4 If Á  < , then
º ²!³ ²!³ “ ²%³» ~ º ²!³ “ ²!³ ²%³»
for all polynomials ²%³  F .
)
Proof. If has the form 19.2 , then by 19.3 ,
(
(
)

(

º! “ !³% » ~ ! Lc 4 5 % c M ~ ~º ²!³ “% » ²19.4 )

~
By linearity, this holds for % replaced by any polynomial ² % ³ . Hence,
applying this to the product gives

º ²!³ ²!³ “ ²%³» ~ º! “ ²!³ ²!³ ²%³»

~ º! “ ²!³´ ²!³ ²%³µ» ~ º ²!³ “ ²!³ ²%³»
(
)
Equation 19.4 shows that applying the linear functional ( !³ is equivalent to
applying the operator ²!³ and then following by evaluation at % ~ .