Determinants and Their Applications in Mathematical Physics

124 4. Particular Determinants

Identities 1.

Vnr=

n

∑

j=1

K

rj

n,^1 ≤r≤n. (4.10.4)

Vnr=

(−1)

n+r

(h+r+n−1)!

(h+r−1)!(r−1)!(n−r)!

, 1 ≤r≤n. (4.10.5)

Vn 1 =

(−1)

n+1

(h+n)!

h!(n−1)!

. (4.10.6)

Vnn=

(h+2n−1)!

(h+n−1)!(n−1)!

. (4.10.7)

K

rs

n =

VnrVns

h+r+s− 1

, 1 ≤r, s≤n. (4.10.8)

K

r 1

n =

VnrVn 1

h+r

. (4.10.9)

K

nn

n

=

Kn− 1

Kn

=

V

2

nn

h+2n− 1

. (4.10.10)

K

rs

n

=

(h+r)(h+s)K

r 1

nK

s 1

n

(h+r+s−1)V

2

n 1

. (4.10.11)

Kn=

(n−1)!

2

(h+n−1)!

2

(h+2n−2)!(h+2n−1)!

Kn− 1. (4.10.12)

Kn=

[1!2!3!···(n−1)!]

2

h!(h+ 1)!···(h+n−1)!

(h+n)!(h+n+ 1)!···(h+2n−1)!

. (4.10.13)

(n−r)Vnr+(h+n+r−1)Vn− 1 ,r=0. (4.10.14)

Kn

n

∏

r=1

Vnr=(−1)

n(n−1)/ 2

. (4.10.15)

Proof. Equation (4.10.4) is a simple expansion ofVnrby elements from

rowr. The following proof of (4.10.5) is a development of one due to Lane.

Perform the row operations

R

′

i

=Ri−Rr, 1 ≤i≤n, i=r,

onKn, that is, subtract rowrfrom each of the other rows. The result is

Kn=|k

′

ij

|n,

where

k

′

rj=krj,

k

′

ij=kij−krj

=

(

r−i

h+r+j− 1

)

kij, 1 ≤i, j≤n, i=r.

After removing the factor (r−i) from each rowi,i=r, and the factor

(h+r+j−1)
− 1
from each columnjand then cancelingKnthe result can