Page 489 - Advanced Linear Algebra

The Umbral Calculus 473

The Umbral Algebra
Let F d ~´%µ denote the algebra of polynomials in a single variable over the
%
complex field. One of the starting points of the umbral calculus is the fact that
any formal power series in can play three different roles: as a formal power
<
series, as a linear functional on and as a linear operator on . Let us first
F
F
explore the connection between formal power series and linear functionals.
Let F i denote the vector space of all linear functionals on . Note that F i is the
F
algebraic dual space of , as defined in Chapter 2. It will be convenient to
F
denote the action of 3 F i on ²%³  F by
º3 “ ²%³»
(This is the “bra-ket” notation of Paul Dirac. ) The vector space operations on F i
then take the form

º3 b 4 “ ²%³» ~ º3 “ ²%³» b º4 “ ²%³»
and
º 3“ ²%³»~ º3“ ²%³»Á  d

Note also that since any linear functional on is uniquely determined by its
F
values on a basis for F the functional 3  FÁ i is uniquely determined by the

values º3 “ % » for ‚ .
Now, any formal series in can be written in the form
<
B

²!³ ~ !
!
~
and we can use this to define a linear functional ²!³ by setting

º ²!³ “ % » ~
for ‚ . In other words, the linear functional ²!³ is defined by

B
²!³ ~ º ²!³ “ % » !
!
~
where the expression ²!³ on the left is just a formal power series. Note in
particular that

º! “ % » ~ ! Á
is the Kronecker delta function. This implies that
where Á
²

³
º! “ ²%³» ~ ² ³

and so is the functional “ th derivative at .” Also, is evaluation at .

!
!