Page 504 - Advanced Linear Algebra

488 Advanced Linear Algebra

Examples of Sheffer Sequences
We can now give some examples of Sheffer sequences. While it is often a
relatively straightforward matter to verify that a given sequence is Sheffer for a
given pair ² ²!³Á ²!³³ , it is quite another matter to find the Sheffer sequence for
a given pair. The umbral calculus provides two formulas for this purpose, one of
which is direct, but requires the usually very difficult computation of the series
² ²!³°!³ c . The other is a recurrence relation that expresses each ²%³ in terms

of previous terms in the Sheffer sequence. Unfortunately, space does not permit
us to discuss these formulas in detail. However, we will discuss the recurrence
formula for associated sequences later in this chapter.

Example 19.4 The sequence ²%³ ~ % is the associated sequence for the delta

series ²!³ ~ ! . The generating function for this sequence is
B &
&!
~ !
[
~
and the binomial identity is the well-known binomial formula

²% b &³ ~ 45 % & c

~
Example 19.5 The lower factorial polynomials
²%³ ~ %²%c ³Ä²%c b ³

form the associated sequence for the forward difference functional
!
²!³ ~ c
discussed in Example 19.2. To see this, we simply compute, using Theorem

19.12. Since ² ³ is defined to be , we have ² ³ ~ Á . Also,
!
² c ³²%³ ~ ²%b ³ c²%³

~ ´²% b ³%²% c ³Ä²% c b ³µ c´%²%c ³Ä²%c b ³µ
~ %²% c ³Ä²% c b ³´²% b ³c²%c b ³µ
~ %²%c ³Ä²%c b ³
~ ²%³ c
The generating function for the lower factorial polynomials is
B
& ² log b ! ³ ~ ²&³ !
[
~