Page 124 - Determinants and Their Applications in Mathematical Physics

4.8 Hankelians 1 109

4.8.5 Turanians
A Hankelian in which a ij = φ i+j−2+r is called a Turanian by Karlin and
o
Szeg¨ and others.
Let

 |φ m+r | n , 0 ≤ m ≤ 2n − 2,

 |φ m | n , r ≤ m ≤ 2n − 2+ r,




···

(n,r)
T = φ r φ n−1+r (4.8.17)
......................





 φ n−1+r
 ··· φ 2n−2+r
 n
C r C r+1 C r+2 ··· C n−1+r .


Theorem 4.28.
(n,r+1)
T T (n,r) (n+1,r−1) (n−1,r+1)
(n,r) = T T .
T T
(n,r−1)
Proof. Denote the determinant by T. Then, each of the Turanian ele-
ments in T is of order n and is a minor of one of the corner elements in
T (n+1,r−1) . Applying the Jacobi identity (Section 3.6),
(n+1,r−1)
T 11 T 1,n+1

(n+1,r−1)
(n+1,r−1) (n+1,r−1)
T =
T T
n+1,1 n+1,n+1

= T (n+1,r−1) T (n+1,r−1)
1,n+1;1,n+1
= T (n+1,r−1) T (n−1,r+1) ,
which proves the theorem.
Let
A n = T (n,0) = |φ i+j−2 | n ,
F n = T (n,1) = |φ i+j−1 | n ,
G n = T (n,2) = |φ i+j | n . (4.8.18)
Then, the particular case of the theorem in which r = 1 can be expressed
in the form
2
A n G n − A n+1 G n−1 = F . (4.8.19)
n
This identity is applied in Section 4.12.2 on generalized geometric series.
Omit the parameter r in T (n,r) and write T n .
Theorem 4.29. For all values of r,
(n)
T 11 T 1,n+1 (n+1)

(n+1)
(n) (n+1) − T n T 1n;1,n+1 =0.
T n1 T n,n+1