Page 139 - Determinants and Their Applications in Mathematical Physics

124 4. Particular Determinants

Identities 1.
n

V nr = K , 1 ≤ r ≤ n. (4.10.4)
rj
n
j=1
(−1) n+r (h + r + n − 1)!
V nr = , 1 ≤ r ≤ n. (4.10.5)
(h + r − 1)!(r − 1)!(n − r)!
(−1) n+1 (h + n)!
V n1 = . (4.10.6)
h!(n − 1)!
(h +2n − 1)!
V nn = . (4.10.7)
(h + n − 1)!(n − 1)!
K rs = V nr V ns , 1 ≤ r, s ≤ n. (4.10.8)
h + r + s − 1
n
K r1 = V nr V n1 . (4.10.9)
h + r
n
V 2
K n−1
K nn = = nn . (4.10.10)
h +2n − 1
n
K n
r1
(h + r)(h + s)K K s1
K rs = n n . (4.10.11)
(h + r + s − 1)V n1
n 2
2
(n − 1)! (h + n − 1)! 2
K n = K n−1 . (4.10.12)
(h +2n − 2)!(h +2n − 1)!
2
[1!2!3! ··· (n − 1)!] h!(h + 1)! ··· (h + n − 1)!
K n = . (4.10.13)
(h + n)!(h + n + 1)! ··· (h +2n − 1)!
(n − r)V nr +(h + n + r − 1)V n−1,r =0. (4.10.14)
n
n(n−1)/2
V nr =(−1) . (4.10.15)
K n
r=1
Proof. Equation (4.10.4) is a simple expansion of V nr by elements from
row r. The following proof of (4.10.5) is a development of one due to Lane.
Perform the row operations

R = R i − R r , 1 ≤ i ≤ n, i = r,
i
on K n , that is, subtract row r from each of the other rows. The result is
K n = |k | n ,

ij
where
k = k rj ,
rj

k = k ij − k rj
ij
r − i

= k ij , 1 ≤ i, j ≤ n, i = r.
h + r + j − 1
After removing the factor (r − i) from each row i, i = r, and the factor
(h + r + j − 1) −1 from each column j and then canceling K n the result can