Determinants and Their Applications in Mathematical Physics

4.10 Henkelians 3 123

Prove that

D

r

(E)=(−1)

r+1

r!

n

∑

i=2

Si− 2 Eir

=(−1)

r+1

r!

∣

∣C

1 C 2 ···Cr− 1 KCr+1···Cn

∣

∣

n

4.10 Henkelians 3

4.10.1 The Generalized Hilbert Determinant........

The generalized Hilbert determinantKnis defined as

Kn=Kn(h)=|kij|n,

where

kij=

1

h+i+j− 1

,h=1−i−j, 1 ≤i, j≤n. (4.10.1)

In some detail,

Kn=

∣

∣

∣

∣

∣

∣

∣

∣

1

h+1

1

h+2

···

1

h+n

1

h+2

1

h+3

···

1

h+n+1

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

1

h+n

1

h+n+1

···

1

h+2n− 1

∣

∣

∣

∣ ∣ ∣ ∣ ∣ n

. (4.10.2)

Knis of fundamental importance in the evaluation of a number of de-

terminants, not necessarily Hankelians, whose elements are related tokij.

The values of such determinants and their cofactors can, in some cases,

be simplified by expressing them in terms ofKnand its cofactors. The

given restrictions onhare the only restrictions onhwhich may therefore

be regarded as a continuous variable. All formulas inhgiven below on the

assumption thathis 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).

LetVnr=Vnr(h) denote a determinantal ratio (not a scaled cofactor)

defined as

Vnr=

1

Kn

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

1

h+1

1

h+2

···

1

h+n

1

h+2

1

h+3

···

1

h+n+1

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

11 ··· 1

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

1

h+n

1

h+n+1

···

1

h+2n− 1

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

rowr, (4.10.3)

where every element in rowris 1 and all the other elements are identical

with the corresponding elements inKn. The following notes begin with the

evaluation ofVnrand end with the evaluation ofKnand its scaled cofactor

K
rs
n

.