Page 110 - Determinants and Their Applications in Mathematical Physics

4.6 Hessenbergians 95

b r = a r , r > 1.

B n (0) = A n ,
(n) (n)
B (0) = A . (4.6.11)
ij ij
Theorem 4.23.
a. B = −nB n−1 .

n
n
(n)
b. A = nA n−1 .
rr
r=1
n
n
c. B n = A r (−x) n−r .
r
r=0
Proof.
B 1 = −x + A 1 ,
2
B 2 = x − 2A 1 x + A 2 ,
3
2
B 3 = −x +3A 1 x − 3A 2 x + A 3 , (4.6.12)
etc., which are Appell polynomials (Appendix A.4) so that (a) is valid for
small values of n. Assume that
B = −rB r−1 , 2 ≤ r ≤ n − 1,

r
and apply the method of induction.
From (4.6.10),
n−2

B n =(n − 1)! a n−r B r +(a 1 − x)B n−1 ,
r!
r=0
n−2
a n−r rB r−1

B = −(n − 1)! − (n − 1)(a 1 − x)B n−2 − B n−1

r!
n
r=1
n−2
a n−r B r−1

= −(n − 1)! − (n − 1)(a 1 − x)B n−2 − B n−1
(r − 1)!
r=1
n−3

= −(n − 1)! a n−1−r B r − (n − 1)(a 1 − x)B n−2 − B n−1
r!
r=0
n−2

= −(n − 1)! b n−1−r B r − B n−1
r!
r=0
= −(n − 1)B n−1 − B n−1
= −nB n−1 ,
which proves (a).