Determinants and Their Applications in Mathematical Physics

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.