Page 502 - Advanced Linear Algebra

486 Advanced Linear Algebra

Theorem 19.13 Let ²%³ be a Sheffer sequence for ² ²!³Á ²!³³ and let ²%³

be associated with ²!³ . Then for any ²!³ we have

²!³ ²%³ ~ 45 º ²!³ “ ²%³» c ²%³

~
Proof. By the expansion theorem
B º ²!³ “ ²%³»

²!³ ~ ²!³ ²!³
!
~
we have
B º ²!³ “ ²%³»

²!³ ²%³ ~ ! ²!³ ²!³ ²%³

~
B º ²!³ “ ²%³»
~ ² ³ c ² % ³

!
~
which is the desired formula.
Theorem 19.14
1 )(The binomial identity ) A sequence ²%³ is the associated sequence for a

delta series ²!³ if and only if it is of binomial type , that is, if and only if it
satisfies the identity

²% b &³ ~ 45 ²&³ c ²%³

~
for all & d .
2 )(The Sheffer identity ) A sequence ²%³ is Sheffer for ² ²!³Á ²!³³ for

some invertible ²!³ if and only if

²% b &³ ~ 45 ²&³ c ²%³

~
for all & d , where ²%³ is the associated sequence for ²!³ .

)
Proof. To prove part 1 , if ²%³ is an associated sequence, then taking

²!³ ~ in Theorem 19.13 gives the binomial identity. Conversely, suppose
&!
that the sequence ²%³ is of binomial type. We will use the operator

characterization to show that ²%³ is an associated sequence. Taking

%~& ~ we have for ~ ,
² ³ ~ ² ³ ² ³

and so ² ³ ~ . Also,