Determinants and Their Applications in Mathematical Physics

138 4. Particular Determinants

v-Numbers satisfy the identities

n

∑

k=1

vnk

i+k− 1

=1, 1 ≤i≤n, (4.11.3)

vni

n

∑

k=1

vnk

(i+k−1)(k+j−1)

=δij, (4.11.4)

vni

n+i− 1

=−

vn− 1 ,i

n−i

, (4.11.5)

n

∑

i=1

vni=n

2

, (4.11.6)

and are related toKnand its scaled cofactors by

K

ij

n

=

vnivnj

i+j− 1

, (4.11.7)

Kn

n

∏

i=1

vni=(−1)

n(n−1)/ 2

. (4.11.8)

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

Bn= detBn,

where

Bn=[bij]n,

bij=

x

2(i+j−1)

−t

2

i+j− 1

, (4.11.9)

and to the function

Gn=

n

∑

j=1

(x

2 j− 1

+t)Bnj, (4.11.10)

which can be expressed as the determinant|gij|nwhose first (n−1) rows

are identical to the first (n−1) rows ofBn. The elements in the last row

are given by

gnj=x

2 j− 1

+t, 1 ≤j≤n.

The analysis employs both matrix and determinantal methods.

Define five matricesKn,Qn,Sn,Hn, andHnas follows:

Kn=

[

1

i+j− 1

]

n

, (4.11.11)