Determinants and Their Applications in Mathematical Physics

94 4. Particular Determinants

Also, adjusting the dummy variable inS 1 and referring to (4.6.4) with

n→n−1,

S 1 =

n− 1

∑

i=0

(−1)

i

(i+1+a)φiψn− 1 −i

=

n− 1

∑

i=1

(−1)

i

iφiψn− 1 −i+(1+a)

n− 1

∑

i=0

(−1)

i

φiψn− 1 −i

=−S 3.

Hence,ψ

′

n=(a+2−n)Fψn−^1 , which is equivalent to the stated result.

Note that ifφ

′

m=(m−1)φm−^1 , thenψ

′

m=−(m−1)ψm−^1. 

4.6.3 A Hessenberg–Appell Characteristic Polynomial

Let

An=|aij|n,

where

aij=

{

aj−i+1,j≥i,

−j, j=i−1,

0 , otherwise.

In some detail,

An=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

∣

∣

∣

∣

a 1 a 2 a 3 a 4 ··· an− 1 an

− 1 a 1 a 2 a 3 ··· an− 2 an− 1

− 2 a 1 a 2 ··· ··· ···

− 3 a 1 ··· ··· ···

··· ··· ···

a 1 a 2

−(n−1) a 1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ n

. (4.6.7)

Applying the recurrence relation in Theorem 4.20,

An=(n−1)!

n− 1

∑

r=0

an−rAr

r!

,n≥ 1 ,A 0 =1. (4.6.8)

LetBn(x) denote the characteristic polynomial of the matrixAn:

Bn=

∣

∣

An−xI

∣

∣

. (4.6.9)

This determinant satisfies the recurrence relation

Bn=(n−1)!

n− 1

∑

r=0

bn−rBr

r!

,n≥ 1 ,B 0 =1, (4.6.10)

where

b 1 =a 1 −x,