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)