Page 493 - Advanced Linear Algebra

The Umbral Calculus 477

)
!
2 The forward difference functional is the delta functional c ,
satisfying
!
º c “ ²%³» ~ ² ³ c ² ³
)
!
3 The Abel functional is the delta functional e , satisfying
!
Z
º!e ! “ ²%³» ~ ² ³
)
4 The invertible functional ² c !³ c satisfies
B
º² c !³ c “ ²%³» ~ ²"³ c" "

as can be seen by setting ²%³ ~ % and expanding the expression
² c !³ .
c
5 To determine the linear functional satisfying
)

º ²!³ “ ²%³» ~ ²"³ "

we observe that
B º ²!³ “ % » B b ! c

!

²!³ ~ ! ~ ! ~
~ ~ ²! b ³ [ !
!
The inverse !°² c ³ of this functional is associated with the Bernoulli
polynomials, which play a very important role in mathematics and its
applications. In fact, the numbers
!
)~ L % c M
!
c
are known as the Bernoulli numbers .
Formal Power Series as Linear Operators
We now turn to the connection between formal power series and linear
operators on . Let us denote the th derivative operator on by . Thus,

F

F
!

³

²
! ²%³ ~ ²%³
We can then extend this to formal series in ,
!
B

²!³ ~ ! ( 19.2)
!
~