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

′′

)

2

=y

′

(

y

x

)′

(

y+4n

2

1+x

)′

,

wherenis a positive integer, is satisfied by the function

y=4(c−1)xA

11

n,

whereA

11

n is a scaled cofactor of the HankelianAn=|aij|nin which

aij=

x

i+j− 1

+(−1)

i+j

c

i+j− 1

andcis an arbitrary constant. The solution is clearly defined whenn≥ 2

but can be made valid whenn=1by adopting the conventionA 11 =1so

thatA

11

=(x+c)

− 1

.

Proof. Using Hankelian notation (Section 4.8),

A=|φm|n, 0 ≤m≤ 2 n− 2 ,

where

φm=

x

m+1

+(−1)

m

c

m+1

. (6.3.1)

Let

P=|ψm|n,

where

ψm=(1+x)

−m− 1

φm.

Then,

ψ

′

m=mF ψm−^1

(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

′

0

P 11

=

(1−c)P 11

(1 +x)

2

. (6.3.3)