Page 487 - Advanced Linear Algebra

Chapter 19

The Umbral Calculus

In this chapter, we give a brief introduction to an area called the umbral
calculus. This is a linear-algebraic theory used to study certain types of
polynomial functions that play an important role in applied mathematics. We
give only a brief introduction to the subject, emphasizing the algebraic aspects
rather than the applications. For more on the umbral calculus, may we suggest
The Umbral Calculus, by Roman 1984 ?
´
µ
One bit of notation: The lower factorial numbers are defined by
² ³ ~ ² c ³Ä² c b ³

Formal Power Series

We begin with a few remarks concerning formal power series. Let denote the
<
algebra of formal power series in the variable , with complex coefficients.
!
Thus, is the set of all formal sums of the form
<
B
²!³ ~ ! (19.1 )

~
the complex numbers . Addition and multiplication are purely
where  d ( )

formal:
B B B
! b ! ~ ² b ³!

~ ~ ~
and
B B B
4 5 4 5 ! ! 4 c 5 ~ !

~ ~ ~ ~
The order ² ³ of is the smallest exponent of that appears with a nonzero

!
coefficient. The order of the zero series is defined to be bB . Note that a series