Page 197 - Determinants and Their Applications in Mathematical Physics

182 5. Further Determinant Theory

Let H ∗ and K ∗ denote the determinants obtained from H n−1 and
n−1 n−1
K n−1 , respectively, by again replacing each φ r by (φ r − xφ r+1 ) and by
replacing each ψ r by (ψ r − xψ r+1 ). In the notation of the second and
fourth lines of (5.2.6),

H ∗ = φ m − xφ m+1 , 1 ≤ m ≤ 2n − 3,
n−1

n

∗
K n−1 = ψ m − xψ m+1 , 1 ≤ m ≤ 2n − 3. (5.2.18)

n
Lemma 5.4.
n

∗
a. H (n) n−i = H n−1 ,
x
in
i=1
n

x
b. K (n) n−i = K ∗ n−1 .
in
i=1
Proof of (a).
n−1
φ 1 φ 2 ··· φ n−1 x
n−2
φ 2 φ 3 ··· x
n
φ n
x
H (n) n−i = ............................... .

in

i=1 φ n−1 φ n ··· φ 2n−3 x

φ n+1 ··· φ 2n−2 1
φ n
n
The result follows by eliminating the x’s from the last column by means of
the row operations:

R = R i − xR i+1 , 1 ≤ i ≤ n − 1.
i
Part (b) is proved in a similar manner.
Lemma 5.5.
n (n) (n)
(n)

(−1) i+1 Pf = H K , 1 ≤ i ≤ 2n − 1.
i jn i−j+1,n
j=1
(n)
Since K mn =0 when m< 1 and when m>n, the true upper limit in the
sum is i, but it is convenient to retain n in 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,
2n−1
n
n
i+1 (n) 2n−i−1
x
x
(−1) Pf x = H (n) n−i K (n) n−s
sn
i in
i=1 i=1 s=1
 

n n s = i − j +1
(n)
=  H K (n) 2n−j−s 
x
sn
jn
j=1 s=1 s → i