Determinants and Their Applications in Mathematical Physics

4.6 Hessenbergians 93

In the next theorem,φmandψmare functions ofx.

Theorem 4.22. If

φ

′

m

=(m+a)Fφm− 1 ,F=F(x),

then

ψ

′

m

=(a+2−m)Fψm− 1.

Proof. It follows from (4.6.4) that

ψn=

n

∑

i=1

(−1)

i+1

φiψn−i. (4.6.6)

It may be verified by elementary methods that

ψ

′

1

=(a+1)Fψ 0 ,

ψ

′

2 =aF ψ^1 ,

ψ

′

3 =(a−1)Fψ 2 ,

etc., so that the theorem is known to be true for small values ofm. Assume

it to be true for 1≤m≤n−1 and apply the method of induction.

Differentiating (4.6.6),

ψ

′

n=

n

∑

i=1

(−1)

i+1

(φ

′

iψn−i+φiψ

′

n−i)

=F

n

∑

i=1

(−1)

i+1

[(i+a)φi− 1 ψn−i+(a+2−n+i)φiψn− 1 −i]

=F(S 1 +S 2 +S 3 ),

where

S 1 =

n

∑

i=1

(−1)

i+1

(i+a)φi− 1 ψn−i,

S 2 =(a+2−n)

n

∑

i=1

(−1)

i+1

φiψn− 1 −i,

S 3 =

n

∑

i=1

(−1)

i+1

iφiψn− 1 −i.

Since thei=nterms inS 2 andS 3 are zero, the upper limits in these sums

can be reduced to (n−1). It follows that

S 2 =(a+2−n)ψn− 1.