Page 138 - Determinants and Their Applications in Mathematical Physics

4.10 Henkelians 3 123

Prove that
n

D (E)=(−1) r+1 r! S i−2 E ir
r
i=2

=(−1) r+1
r! C 1 C 2 ··· C r−1 KC r+1 ··· C n .
n
4.10 Henkelians 3

4.10.1 The Generalized Hilbert Determinant
The generalized Hilbert determinant K n is defined as
K n = K n (h)= |k ij | n ,

where
1
k ij = , h =1 − i − j, 1 ≤ i, j ≤ n. (4.10.1)
h + i + j − 1
In some detail,
1 1 1

h+1 h+2 ··· h+n
1 1 1
···
K n = h+2 h+3 h+n+1 . (4.10.2)
...........................

1 1 1
···
h+n+1
h+n h+2n−1 n
K n is of fundamental importance in the evaluation of a number of de-
terminants, not necessarily Hankelians, whose elements are related to k ij .
The values of such determinants and their cofactors can, in some cases,
be simplified by expressing them in terms of K n and its cofactors. The
given restrictions on h are the only restrictions on h which may therefore
be regarded as a continuous variable. All formulas in h given below on the
assumption that h is zero, a positive integer, or a permitted negative in-
teger can be modified to include other permitted values by replacing, for
example, (h + n)! by Γ(h + n + 1).
Let V nr = V nr (h) denote a determinantal ratio (not a scaled cofactor)
defined as
1 1 1

···
h+1 h+2 h+n
1 1 1
···
h+2 h+3 h+n+1
1 ...........................
V nr = row r, (4.10.3)
1 1 ··· 1
K n

...........................

1 1 1
···
h+n+1
h+n h+2n−1 n
where every element in row r is 1 and all the other elements are identical
with the corresponding elements in K n . The following notes begin with the
evaluation of V nr and end with the evaluation of K n and its scaled cofactor
K .
rs
n