Page 506 - Advanced Linear Algebra

490 Advanced Linear Algebra

We ask the reader to show that ²%³ is the Appell sequence for ²!³ if and only

c
if ²%³ ~ ²!³ % . Using this fact, we get

% ~
/²%³ ~ c! ° ²c ³ ² ³ % c

‚ [
The generating function for the Hermite polynomials is
B /²&³

&!c! ° ~ !
[
~
and the Sheffer identity is

/²% b &³ ~ 45 / ²%³& c

~
We should remark that the Hermite polynomials, as defined in the literature,
often differ from our definition by a multiplicative constant.

Example 19.8 The well-known and important Laguerre polynomials 3 ²³ ² % ³
of order form the Sheffer sequence for the pair

!

²!³ ~ ² c !³ cc Á ²!³ ~
!c
It is possible to show although we will not do so here that
²
³
[ b

3 ²³ ² % ³ ~ 4 ² c5 % ³

~ [ c
The generating function of the Laguerre polynomials is

B ²³
&!°²!c ³ 3 ² % ³
~ !
² c !³ b [
~
As with the Hermite polynomials, some definitions of the Laguerre polynomials
differ by a multiplicative constant.
We presume that the few examples we have given here indicate that the umbral
calculus applies to a significant range of important polynomial sequences. In
Roman 1984 , we discuss approximately 30 different sequences of polynomials
µ
´
(
)
that are or are closely related to Sheffer sequences.
Umbral Operators and Umbral Shifts
We have now established the basic framework of the umbral calculus. As we
have seen, the umbral algebra plays three roles: as the algebra of formal power
series in a single variable, as the algebra of all linear functionals on and as the
F