Determinants and Their Applications in Mathematical Physics

4.6 Hessenbergians 95

br=ar,r> 1.

Bn(0) =An,

B

(n)

ij

(0) =A

(n)

ij

. (4.6.11)

Theorem 4.23.

a.B

′

n

=−nBn− 1.

b.

n

∑

r=1

A

(n)

rr =nAn−^1.

c. Bn=

n

∑

r=0

(

n

r

)

Ar(−x)

n−r

.

Proof.

B 1 =−x+A 1 ,

B 2 =x

2

− 2 A 1 x+A 2 ,

B 3 =−x

3

+3A 1 x

2

− 3 A 2 x+A 3 , (4.6.12)

etc., which are Appell polynomials (Appendix A.4) so that (a) is valid for

small values ofn. Assume that

B

′

r=−rBr−^1 ,^2 ≤r≤n−^1 ,

and apply the method of induction.

From (4.6.10),

Bn=(n−1)!

n− 2

∑

r=0

an−rBr

r!

+(a 1 −x)Bn− 1 ,

B

′

n=−(n−1)!

n− 2

∑

r=1

an−rrBr− 1

r!

−(n−1)(a 1 −x)Bn− 2 −Bn− 1

=−(n−1)!

n− 2

∑

r=1

an−rBr− 1

(r−1)!

−(n−1)(a 1 −x)Bn− 2 −Bn− 1

=−(n−1)!

n− 3

∑

r=0

an− 1 −rBr

r!

−(n−1)(a 1 −x)Bn− 2 −Bn− 1

=−(n−1)!

n− 2

∑

r=0

bn− 1 −rBr

r!

−Bn− 1

=−(n−1)Bn− 1 −Bn− 1

=−nBn− 1 ,

which proves (a).