Page 109 - Determinants and Their Applications in Mathematical Physics

94 4. Particular Determinants

Also, adjusting the dummy variable in S 1 and referring to (4.6.4) with
n → n − 1,
n−1

S 1 = (−1) (i +1+ a)φ i ψ n−1−i
i
i=0
n−1 n−1

= (−1) iφ i ψ n−1−i +(1+ a) (−1) φ i ψ n−1−i
i
i
i=1 i=0
= −S 3 .
Hence, ψ =(a +2 − n)Fψ n−1 , which is equivalent to the stated result.

n
Note that if φ =(m − 1)φ m−1 , then ψ = −(m − 1)ψ m−1 .
m m
4.6.3 A Hessenberg–Appell Characteristic Polynomial
Let
A n = |a ij | n ,
where
a j−i+1 ,j ≥ i,

a ij = −j, j = i − 1,
0, otherwise.
In some detail,

a 1 a 2 a 3 a 4 ··· a n−1 a n
−1 a 1 a 2 a 3 ··· a n−2 a n−1

−2 ··· ···

a 1 a 2 ···
··· . (4.6.7)

−3 a 1 ··· ···
A n =

··· ···
···

a 1
a 2
−(n − 1) a 1

n
Applying the recurrence relation in Theorem 4.20,
n−1

A n =(n − 1)! a n−r A r , n ≥ 1, A 0 =1. (4.6.8)
r!
r=0
Let B n (x) denote the characteristic polynomial of the matrix A n :

B n = A n − xI . (4.6.9)

This determinant satisfies the recurrence relation
n−1

B n =(n − 1)! b n−r B r , n ≥ 1, B 0 =1, (4.6.10)
r!
r=0
where
b 1 = a 1 − x,