Determinants and Their Applications in Mathematical Physics

104 4. Particular Determinants

=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

ff

′

f

′′

··· f

(n−1)

f

′

f

′′

f

′′′

··· ···

f

′′

f

′′′

f

(4)

··· ···

··· ··· ··· ··· ···

f

(n−1)

··· ··· ··· f

(2n−2)

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

. (4.7.19)

Then, the rows and columns satisfy the relation

R

′

i

=Ri+1,

C

′

j

=Cj+1, (4.7.20)

which contrasts with the simple Wronskian defined above in which only one

of these relations is valid. Determinants of this form are known as two-way

or double Wronskians. They are also Hankelians. A more general two-way

Wronskian is the determinant

Wn=

∣

∣Di−^1

x

D

j− 1

y

(f)

∣

∣

n

(4.7.21)

in which

Dx(Ri)=Ri+1,

Dy(Cj)=Cj+1. (4.7.22)

Two-way Wronskians appear in Section 6.5 on Toda equations.

Exercise.LetAandBdenote Wronskians of ordernwhose columns are

defined as follows:

InA,

C 1 =

[

1 xx

2

···x

n− 1

]

, Cj=Dx(Cj− 1 ).

InB,

C 1 =

[

1 yy

2

···y

n− 1

]

, Cj=Dy(Cj− 1 ).

Now, letEdenote the hybrid determinant of ordernwhose firstrcolumns

are identical with the firstrcolumns ofAand whose lastscolumns are

identical with the firstscolumns ofB, wherer+s=n. Prove that

E=

[

0! 1! 2!···(r−1)!

][

0! 1! 2!···(s−1)!

]

(y−x)

rs

. (Corduneanu)

4.8 Hankelians 1

4.8.1 Definition and theφmNotation

A Hankel determinantAnis defined as

An=|aij|n,

where

aij=f(i+j). (4.8.1)