Page 141 - Determinants and Their Applications in Mathematical Physics

126 4. Particular Determinants

(−1) r+1 n(r + n − 1)!
r1
4. a. K (0) = .
(r − 1)!r!(n − r)!
n

(−1) r+s rs n − 1 n − 1 r + n − 1 s + n − 1
b. K (0) = .
rs
r + s − 1
n r − 1 s − 1 r s
[1!2!3! ··· (n − 1)!] 3
c. K n (0) = .
n!(n + 1)!(n + 2)! ··· (2n − 1)!
5.

1 1
=2 n
2 2i +2j − 1
K n
n
n−1
2 2 (2r + 1)!(r + n)!
=2 2n [1!2!3! ··· (n − 1)!] .
r!(2r +2n + 1)!
r=0
[Apply the Legendre duplication formula in Appendix A.1].
6. By choosing h suitably, evaluate |1/(2i +2j − 3)| n .
The next set of identities are of a different nature. The parameter n is
omitted from V nr , K , and so forth.
ij
n
Identities 2.
K sj
= δ rs , 1 ≤ r ≤ n. (4.10.16)
h + r + j − 1
j
V j
=1, 1 ≤ r ≤ n. (4.10.17)
h + r + j − 1
j

= δ rs , 1 ≤ r, s ≤ n.(4.10.18)
V j
(h + r + j − 1)(h + s + j − 1)
j V r
jK 1j
= V 1 − hδ r1 , 1 ≤ r ≤ n. (4.10.19)
h + r + j − 1
j

V j = K ij = n(n + h). (4.10.20)
j i j
2
jK 1j =(n + nh − h)V 1 . (4.10.21)
j
Proof. Equation (4.10.16) is simply the identity

k rj K sj = δ rs .
j
To prove (4.10.17), apply (4.10.9) with r → j and (4.10.4): and (4.10.12),
(h + j)K j1
V 1 V j =
h + r + j − 1 h + r + j − 1
j j

r − 1 j1
= 1 − K
h + r + j − 1
j