Page 522 - Advanced Linear Algebra

506 Advanced Linear Algebra

Finally, it is possible to generalize the classical umbral calculus that we have
described in this chapter to provide a context for studying polynomial sequences
such as those of the names Gegenbauer, Chebyshev and Jacobi. Also, there is a
(
q-version of the umbral calculus that involves the q-binomial coefficients also
known as the Gaussian coefficients)
² c ³ Ä ² c ³
45 ~

² c ³Ä² c ³² c ³Ä² c c ³
in place of the binomial coefficients. There is also a logarithmic version of the
umbral calculus, which studies the harmonic logarithms and sequences of
logarithmic type. For more on these topics, please see [103], [106] and [107].
Exercises

1. Prove that ² ³ ~ ² ³ b ² ³ , for any Á  < .
2. Prove that ² b ³ ‚ min¸ ² ³Á ² ³¹ , for any Á  < .
3. Show that any delta series has a compositional inverse.
4. Show that for any delta series , the sequence is a pseudobasis.

5. Prove that is a derivation.
C !
6. Show that  < is a delta functional if and only if º “ » ~ and
º “ %» £ .
7. Show that  < is invertible if and only if º “ » £ .
8. Show that º ² !³ “ ²%³» ~ º ²!³ “ ² %³» for any a  d ,  < and
 F.
!
Z
º!
9. Show that e “ ²%³» ~ ² a for any polynomial ²%³  F .
³
10. Show that ~ in < if and only if ~ as linear functionals, which
holds if and only if ~ as linear operators.
11. Prove that if ²%³ is Sheffer for ² ²!³Á ²!³³ , then ²!³ ²%³ ~ c ²%³ .

Hint: Apply the functionals ²!³ ²!³ to both sides.
12. Verify that the Abel polynomials form the associated sequence for the Abel
functional.
13. Show that a sequence ²%³ is the Appell sequence for ²!³ if and only if

c
²%³ ~ ²!³ % .

14. If is a delta series, show that the adjoint d of the umbral operator is a

vector space isomorphism of .
<
15. Prove that if is an automorphism of the umbral algebra, then preserves
;
;
order, that is, ²; ²!³³ ~ ² ²!³³ . In particular, is continuous.
;
16. Show that an umbral operator maps associated sequences to associated
sequences.
17. Let ²%³ and ²%³ be associated sequences. Define a linear operator by

¢ ²%³ ¦ ²%³. Show that is an umbral operator.
18. Prove that if C and C are surjective derivations on < , then
C ²!³ ~ ´C ²!³µ .
c