Page 498 - Advanced Linear Algebra

482 Advanced Linear Algebra

Note that the sequence ²%³ is Sheffer for ² ²!³Á ²!³³ if and only if

º ²!³ ²!³ “ ²%³» ~ [ Á
which is equivalent to

º ²!³ “ ²!³ ²%³» ~ [ Á
which, in turn, is equivalent to saying that the sequence ²%³ ~ ²!³ ²%³ is

the associated sequence for ²!³ .
Theorem 19.8 The sequence ²%³ is Sheffer for ² ²!³Á ²!³³ if and only if the

sequence ²%³ ~ ²!³ ²%³ is the associated sequence for ²!³ .

Before considering examples, we wish to describe several characterizations of
Sheffer sequences. First, we require a key result.

(
Theorem 19.9 The expansion theorems) Let ²%³ be Sheffer for ² ²!³Á ²!³³ .

)
1 For any  < ,
B º ²!³ “ ²%³»

²!³ ~ ²!³ ²!³
~ !
)
2 For any  F ,

º ²!³ ²!³ “ ²%³
²%³ ~ ²%³

!
‚
Proof. Part 1 follows from Theorem 19.2, since
)
B º ²!³ “ B º ²!³ “ ²%³»
²%³»

L ²!³ ²!³ c M ! ²%³ ~ Á
! !
~ ~
~ º ²!³ “ ²%³»
)
Part 2 follows in a similar way from Theorem 19.2.
We can now begin our characterization of Sheffer sequences, starting with the
generating function. The idea of a generating function is quite simple. If ²%³ is

a sequence of polynomials, we may define a formal power series of the form
B ²%³
²!Á %³ ~ !
!
~
This is referred to as the (exponential generating function for the sequence
)
²%³. The term exponential refers to the presence of ( ! in this series. When

this is not present, we have an ordinary generating function. Since the series is
³
(
a formal one, knowing ²!Á %³ is equivalent in theory, if not always in practice)