Page 108 - Determinants and Their Applications in Mathematical Physics

4.6 Hessenbergians 93

In the next theorem, φ m and ψ m are functions of x.
Theorem 4.22. If

φ =(m + a)Fφ m−1 , F = F(x),
m
then
ψ =(a +2 − m)Fψ m−1 .
m
Proof. It follows from (4.6.4) that

n
i+1
ψ n = (−1) φ i ψ n−i . (4.6.6)
i=1
It may be verified by elementary methods that

ψ =(a +1)Fψ 0 ,
1

ψ = aFψ 1 ,
2
ψ =(a − 1)Fψ 2 ,

3
etc., so that the theorem is known to be true for small values of m. Assume
it to be true for 1 ≤ m ≤ n − 1 and apply the method of induction.
Differentiating (4.6.6),
n
i+1
ψ = (−1) (φ ψ n−i + φ i ψ )

n i n−i
i=1
n
i+1
= F (−1) [(i + a)φ i−1 ψ n−i +(a +2 − n + i)φ i ψ n−1−i ]
i=1
= F(S 1 + S 2 + S 3 ),
where
n
i+1
S 1 = (−1) (i + a)φ i−1 ψ n−i ,
i=1
n
i+1
S 2 =(a +2 − n) (−1) φ i ψ n−1−i ,
i=1
n
i+1
S 3 = (−1) iφ i ψ n−1−i .
i=1
Since the i = n terms in S 2 and S 3 are zero, the upper limits in these sums
can be reduced to (n − 1). It follows that
S 2 =(a +2 − n)ψ n−1 .