Page 519 - Advanced Linear Algebra
P. 519
The Umbral Calculus 503
Proof. We have
º ²!³ d ²!³ ²%³»~º´ d ²!³µ ²!³ ²%³»
~ º d ´ ²!³ ²!³µ c ²!³ d ²!³ ²%³»
~ º d ´ ²!³ ²!³µ ²%³» c º ²!³ c ²!³ ²%³»
~º ²!³ ²!³ ²%³» c º c ²!³ ²!³ ²%³»
d
~ º ²!³ ²!³ ²%³» c º ²!³ ²!³ ²%³»
~ º ²!³ ²!³ ²%³» c º ²!³ ²!³ ²%³»
from which the result follows.
%
If ²!³ ~ ! , then is multiplication by and d is the derivative with respect to
! and so the previous result becomes
Z
²!³ ~ ²!³% c % ²!³
as operators on . The right side of this is called the Pincherle derivative of
F
(
the operator ²!³ . See [104].)
Sheffer Shifts
Recall that the linear map
Á ´ ²%³µ ~ b ²%³
where ²%³ is Sheffer for ² ²!³Á ²!³³ is called a Sheffer shift. If ²%³ is
associated with ²!³ , then ²!³ ²%³ ~ ²%³ and so
c c
²!³ b ²%³ ~ Á ´ ²!³ ²%³µ
and so
c Á ~ ²!³ ²!³
From Theorem 19.26, the recurrence formula and the chain rule, we have
c Á ~ ²!³ ²!³
c
~ ²!³´ ²!³ c d ²!³µ
~ c ² ! ³ C c ² ! ³
~ c ² ! ³ C c ² ! ³
~ c ² ! ³ C c ! C ! ² ! ³
Z
c
c
Z
Z
~ %´ ²!³µ c c ²!³´ ²!³µ ²!³
Z
²!³
~% c ?
>
Z
²!³ ²!³
We have proved the following.
