Page 520 - Advanced Linear Algebra

504 Advanced Linear Algebra

be a Sheffer shift. Then
Theorem 19.27 Let Á
Z
²!³
<
1) Á ~% c =
Z
²!³ ²!³
Z
²!³
2) ² b % ³ ~ % c< = ² % ³
Z
²!³ ²!³
The Transfer Formulas
We conclude with a pair of formulas for the computation of associated
sequences.
(
Theorem 19.28 The transfer formulas ) Let ²%³ be the associated sequence

for ²!³ . Then
c c
1) ²%³ ~ ²!³4 Z ²!³ 5 %

!
c
2) ²%³ ~ %4 ²!³ 5 % c

!
)
)
Proof. First we show that 1 and 2 are equivalent. Write ²!³ ~ ²!³°! . Then
Z
²!³ ²!³ c c Z c c
% ~ ´! ²!³µ ²!³
%
%
~ ²!³ c Z c c
% b ! ²!³ ²!³
~ ²!³ c Z c c c
%
% b ²!³ ²!³
~ ²!³ c c Z c
% b ´ ²!³
µ %
% c ´ ²!³
~ ²!³ c c %c% ²!³ c µ% c
%
~% ²!³ c c
To prove 1 , we verify the operation conditions for an associated sequence for
)
Z
the sequence ²%³ ~ ²!³ ²!³ c c % . First, when ‚ the fourth equality

above gives

Z

º! “ ²%³» ~ º! “ ²!³ ²!³ c c % »

Z

~º! “ ²!³ c % c ´ ²!³ c µ % c »

~ º ²!³ c “ % » c º´ ²!³ c Z c »
µ “ %

~ º ²!³ c “ % » c º ²!³ c “ % »
~

If ~ , then º!“ ²%³» ~ , and so in general, we have º!“ ²%³» ~ Á as

required.
For the second required condition,
Z
²!³ ²%³ ~ ²!³ ²!³ ²!³ c c %

Z
~ ! ²!³ ²!³ ²!³ c c %
Z
~ ²!³ ²!³ c c % c
~ c ²%³
Thus, ²%³ is the associated sequence for ²!³ .