Page 165 - Determinants and Their Applications in Mathematical Physics

150 4. Particular Determinants

Note that U ij = V ij in general. Since
ψ = −mψ m−1 ,
m
ψ 0 = x + c, (4.11.68)
it follows that

V = V 11

c(1 + x) i+j+1 +(1 − c)x i+j+1
= . (4.11.69)
i + j +1
n−1
Expand U and V as a polynomial in c:
n

U(x, c)= V (x, c)= f r (x)c n−r . (4.11.70)
r=0
However, since
ψ m = y m c + z m ,
where z m is independent of c,

m+1 m+1
(1 + x) − x
y m =(−1) m , (4.11.71)
m +1
y = −my m−1 ,
m
y 0 =1, (4.11.72)
it follows from the first line of (4.11.67) that f 0 , the coefficient of c in V ,
n
is given by

f 0 = |y m | n
= constant. (4.11.73)

n−1

n−1
−1
−1
c V (x, c )= f 0 c + f 1 + f r+1 c ,
r
r=1
where
∂
n−1
−1

f = c D x V (x, c ) c=0 , D x = ∂x ,
1
n−1 −1
= c V 11 (x, c )
c=0

−1 i+j+1 −1
c (1 + x) +(1 − c )x i+j+1
= c n−1
i + j +1

n−1
c=0
= E. (4.11.74)
Furthermore,
∂
−1
−1
D c {c U(x, c )} = D c {c V (x, c )}, D c = ,
n
n
∂c