Determinants and Their Applications in Mathematical Physics

4.11 Hankelians 4 137

in which the column difference is

1

3

(v

3

−u

3

).

Let the determinant of the elements in the firstnrows and the firstn

columns of the matrix be denoted byAn. Prove that

An=

Knn!

3

(2n)!

(u−v)

n(n+1)

2.Define a HankelianBnas follows:

Bn=

∣

∣

∣

∣

φm

(m+ 1)(m+2)

∣

∣

∣

∣

n

, 0 ≤m≤ 2 n− 2 ,

where

φm=

m

∑

r=0

(m+1−r)u

m−r

v

r

Prove that

Bn=

An+1

n!(u−v)

2 n

,

whereAnis defined in Exercise 1.

4.11 Hankelians 4

Throughout this section,Kn=Kn(0), the simple Hilbert determinant.

4.11.1 v-Numbers

The integersvnidefined by

vni=Vni(0) =

(−1)

n+i

(n+i−1)!

(i−1)!

2

(n−i)!

(4.11.1)

=(−1)

n+i

i

(

n− 1

i− 1

)(

n+i− 1

n− 1

)

, 1 ≤i≤n, (4.11.2)

are of particular interest and will be referred to asv-numbers.

A few values of thev-numbersvniare given in the following table:

i

n 123 45

1 1

2 − 26

3 3 − 24 30

4 − 460 − 180 140

5 5 − 120 630 −1120 630