Page 261 - Determinants and Their Applications in Mathematical Physics

246 6. Applications of Determinants in Mathematical Physics

Equation (6.2.25) was solved by Ohta, Kajiwara, Matsukidaira, and
Satsuma in 1993. A brief note on the solutions is given in Section 6.11.

6.3 The Dale Equation

Theorem. The Dale equation, namely
y y +4n
1 2 2
2
(y ) = y ,
x 1+ x
where n is a positive integer, is satisfied by the function
11
y =4(c − 1)xA ,
n
where A 11 is a scaled cofactor of the Hankelian A n = |a ij | n in which
n
x i+j−1 +(−1) i+j c
a ij =
i + j − 1
and c is an arbitrary constant. The solution is clearly defined when n ≥ 2
but can be made valid when n =1 by adopting the convention A 11 =1 so
that A 11 =(x + c) −1 .
Proof. Using Hankelian notation (Section 4.8),
A = |φ m | n , 0 ≤ m ≤ 2n − 2,
where
x m+1 +(−1) c
m
φ m = . (6.3.1)
m +1
Let
P = |ψ m | n ,
where
ψ m =(1 + x) −m−1 φ m .

Then,
ψ = mFψ m−1
m
(the Appell equation), where
F =(1 + x) −2 . (6.3.2)

Hence, by Theorem 4.33 in Section 4.9.1 on Hankelians with Appell
elements,

P = ψ P 11
0
(1 − c)P 11
= . (6.3.3)
(1 + x) 2