Page 152 - Determinants and Their Applications in Mathematical Physics

4.11 Hankelians 4 137

1
3
3
in which the column difference is (v − u ).
3
Let the determinant of the elements in the first n rows and the first n
columns of the matrix be denoted by A n . Prove that
K n n! 3 n(n+1)
A n = (u − v) .
(2n)!
2. Define a Hankelian B n as follows:

B n = φ m , 0 ≤ m ≤ 2n − 2,
(m + 1)(m +2)
n
where
m

v .
φ m = (m +1 − r)u m−r r
r=0
Prove that
A n+1
B n = ,
n!(u − v) 2n
where A n is defined in Exercise 1.

4.11 Hankelians 4

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

4.11.1 v-Numbers

The integers v ni defined by
(−1) n+i (n + i − 1)!
v ni = V ni (0) = (4.11.1)
2
(i − 1)! (n − i)!

n − 1 n + i − 1
=(−1) n+i i , 1 ≤ i ≤ n, (4.11.2)
i − 1 n − 1
are of particular interest and will be referred to as v-numbers.
A few values of the v-numbers v ni are given in the following table:

i
1 2 3 4 5
n
1 1
2 −2 6
3 3 −24 30
4 −4 60 −180 140
5 5 −120 630 −1120 630