Determinants and Their Applications in Mathematical Physics

182 5. Further Determinant Theory

LetH

∗

n− 1

andK

∗

n− 1

denote the determinants obtained fromHn− 1 and

Kn− 1 , respectively, by again replacing eachφrby (φr−xφr+1) and by

replacing eachψrby (ψr−xψr+1). In the notation of the second and

fourth lines of (5.2.6),

H

∗

n− 1 =

∣

∣φ

m−xφm+1

∣

∣

n

, 1 ≤m≤ 2 n− 3 ,

K

∗

n− 1

=

∣

∣ψ

m−xψm+1

∣

∣

n

, 1 ≤m≤ 2 n− 3. (5.2.18)

Lemma 5.4.

a.

n

∑

i=1

H

(n)

in

x

n−i

=H

∗

n− 1

,

b.

n

∑

i=1

K

(n)

in

x

n−i

=K

∗

n− 1

.

Proof of (a).

n

∑

i=1

H

(n)

in

x

n−i

=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

φ 1 φ 2 ··· φn− 1 x

n− 1

φ 2 φ 3 ··· φn x

n− 2

...............................

φn− 1 φn ··· φ 2 n− 3 x

φn φn+1 ··· φ 2 n− 2 1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

.

The result follows by eliminating thex’s from the last column by means of

the row operations:

R

′

i

=Ri−xRi+1, 1 ≤i≤n− 1.

Part (b) is proved in a similar manner. 

Lemma 5.5.

(−1)

i+1

Pf

(n)

i

=

∑n

j=1

H

(n)

jn

K

(n)

i−j+1,n

, 1 ≤i≤ 2 n− 1.

SinceK

(n)

mn=0whenm<^1 and whenm>n, the true upper limit in the

sum isi, but it is convenient to retainnin order to simplify the analysis

involved in its application.

Proof. It follows from the inductive assumption (5.2.10) that

Pf

∗

n− 1 =H

∗

n− 1 K

∗

n− 1. (5.2.19)

Hence, applying Lemmas 5.3 and 5.4,

2 n− 1

∑

i=1

(−1)

i+1

Pf

(n)

i x

2 n−i− 1

=

[

n

∑

i=1

H

(n)

inx

n−i

][

n

∑

s=1

K

(n)

snx

n−s

]

=





n

∑

j=1

n

∑

s=1

H

(n)

jnK

(n)

snx

2 n−j−s





[

s=i−j+1

s→i

]