Page 119 - Determinants and Their Applications in Mathematical Physics

104 4. Particular Determinants
f f f ··· f
(n−1)

f f f ··· ···

f f f ··· ··· . (4.7.19)
(4)
=

··· ··· ··· ··· ···
f ··· ··· ··· f
(n−1) (2n−2)
n
Then, the rows and columns satisfy the relation

R = R i+1 ,
i

C = C j+1 , (4.7.20)
j
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
i−1 j−1
W n = D D (f) (4.7.21)

x y
n
in which
D x (R i )= R i+1 ,
D y (C j )= C j+1 . (4.7.22)
Two-way Wronskians appear in Section 6.5 on Toda equations.
Exercise. Let A and B denote Wronskians of order n whose columns are
defined as follows:
In A,
2 n−1
C 1 = 1 xx ··· x , C j = D x (C j−1 ).
In B,
2 n−1
C 1 = 1 yy ··· y , C j = D y (C j−1 ).
Now, let E denote the hybrid determinant of order n whose first r columns
are identical with the first r columns of A and whose last s columns are
identical with the first s columns of B, where r + s = n. Prove that

E = 0! 1! 2! ··· (r − 1)! 0! 1! 2! ··· (s − 1)! (y − x) . (Corduneanu)
rs
4.8 Hankelians 1

4.8.1 Definition and the φ m Notation
A Hankel determinant A n is defined as
A n = |a ij | n ,
where
a ij = f(i + j). (4.8.1)