Page 499 - Advanced Linear Algebra

The Umbral Calculus 483

to knowing the polynomials ²%³ . Moreover, a knowledge of the generating

function of a sequence of polynomials can often lead to a deeper understanding
of the sequence itself, that might not be otherwise easily accessible. For this
reason, generating functions are studied quite extensively.

For the proofs of the following characterizations, we refer the reader to Roman
´ µ1984 .

(
Theorem 19.10 Generating function)
)
1 The sequence ²%³ is the associated sequence for a delta series ²!³ if and

only if
B ²&³
& ²!³ ~ ! !
~
where ²!³ is the compositional inverse of ²!³ .
)
2 The sequence ²%³ is Sheffer for ² ²!³Á ²!³³ if and only if

& ²!³ B ² & ³
~ !
² ²!³³ !
~
The sum on the right is called the generating function of ²%³ .

Proof. Part 1 is a special case of part 2 . For part 2 , the expression above is
)
)
)
equivalent to
B
² & ³
&!

~ ²!³
²!³ !
~
which is equivalent to
B ²&³
&!

~ ²!³ ²!³
!
~
But if ²%³ is Sheffer for ² ²!³Á ²!³³ , then this is just the expansion theorem

for &! . Conversely, this expression implies that
B ²&³
&!

²&³ ~ º “ ²%³» ~ ! º ²!³ ²!³ “ ²%³»

~
is Sheffer for

and so º ²!³ ²!³ “ ²%³» ~ [ Á , which says that ²%³
² Á ³.
We can now give a representation for Sheffer sequences.
(
Theorem 19.11 Conjugate representation)