Page 153 - Determinants and Their Applications in Mathematical Physics

138 4. Particular Determinants

v-Numbers satisfy the identities
n
v nk
=1, 1 ≤ i ≤ n, (4.11.3)
i + k − 1
k=1
n

= δ ij , (4.11.4)
v nk
(i + k − 1)(k + j − 1)
v ni
k=1
v ni
= − v n−1,i , (4.11.5)
n + i − 1 n − i
n
2
v ni = n , (4.11.6)
i=1
and are related to K n and its scaled cofactors by
K ij = v ni v nj , (4.11.7)
i + j − 1
n
n
n(n−1)/2
v ni =(−1) . (4.11.8)
K n
i=1
The proofs of these identities are left as exercises for the reader.
4.11.2 Some Determinants with Determinantal Factors

This section is devoted to the factorization of the Hankelian
B n = det B n ,
where

B n =[b ij ] n ,
x 2(i+j−1) − t 2
b ij = , (4.11.9)
i + j − 1
and to the function

n
2j−1
G n = (x + t)B nj , (4.11.10)
j=1
which can be expressed as the determinant |g ij | n whose first (n − 1) rows
are identical to the first (n − 1) rows of B n . The elements in the last row
are given by
g nj = x 2j−1 + t, 1 ≤ j ≤ n.
The analysis employs both matrix and determinantal methods.
Define five matrices K n , Q n , S n , H n , and H n as follows:

1
K n = , (4.11.11)
i + j − 1
n