4.8 Hankelians 1 109
4.8.5 Turanians........................
A Hankelian in whichaij=φi+j−2+ris called a Turanian by Karlin and
Szeg ̈o and others.
Let
T
(n,r)
=
|φm+r|n, 0 ≤m≤ 2 n−2,
|φm|n,r≤m≤ 2 n−2+r,
∣ ∣ ∣ ∣ ∣ ∣
φr ··· φn−1+r
......................
φn−1+r ··· φ 2 n−2+r
∣ ∣ ∣ ∣ ∣ ∣
∣ n
∣
CrCr+1Cr+2···Cn−1+r
∣
∣
.
(4.8.17)
Theorem 4.28.
∣
∣
∣
∣
T
(n,r+1)
T
(n,r)
T
(n,r)
T
(n,r−1)
∣
∣
∣
∣
=T
(n+1,r−1)
T
(n− 1 ,r+1)
.
Proof. Denote the determinant byT. Then, each of the Turanian ele-
ments inTis of ordernand is a minor of one of the corner elements in
T
(n+1,r−1)
. Applying the Jacobi identity (Section 3.6),
T=
∣
∣
∣
∣
∣
T
(n+1,r−1)
11
T
(n+1,r−1)
1 ,n+1
T
(n+1,r−1)
n+1, 1 T
(n+1,r−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
An=T
(n,0)
=|φi+j− 2 |n,
Fn=T
(n,1)
=|φi+j− 1 |n,
Gn=T
(n,2)
=|φi+j|n. (4.8.18)
Then, the particular case of the theorem in whichr= 1 can be expressed
in the form
AnGn−An+1Gn− 1 =F
2
n. (4.8.19)
This identity is applied in Section 4.12.2 on generalized geometric series.
Omit the parameterrinT
(n,r)
and writeTn.
Theorem 4.29. For all values ofr,
∣
∣
∣
∣
∣
T
(n)
11
T
(n+1)
1 ,n+1
T
(n)
n 1
T
(n+1)
n,n+1
∣
∣
∣
∣
∣
−TnT
(n+1)
1 n;1,n+1=0.