Page 262 - Determinants and Their Applications in Mathematical Physics

6.3 The Dale Equation 247

Note that the theorem cannot be applied to A directly since φ m does not
satisfy the Appell equation for any F(x).
Using the identity
|t i+j−2 a ij | n = t n(n−1) |a ij | n ,
it is found that
2
P =(1 + x) −n A,
2
−n +1
P 11 =(1 + x) A 11 . (6.3.4)
Hence,

2
(1 + x)A = n A − (c − 1)A 11 . (6.3.5)

Let
r−1
α i = x A , (6.3.6)
ri
r

β i = (−1) A , (6.3.7)
r
ri
r

λ = (−1) α r
r
r
r s−1
= (−1) x A rs
r s
s−1
= x β s , (6.3.8)
s
where r and s =1, 2, 3,...,n in all sums.
Applying double-sum identity (D) in Section 3.4 with f r = r and g s =
s − 1, then (B),
r+s−1
(i + j − 1)A = [x +(−1) r+s c]A A sj
ri
ij
r s
(6.3.9)
= xα i α j + cβ i β j

ij i+j−2 is rj
(A ) = − x A A
r s
= −α i α j . (6.3.10)
Hence,
ij
x(A ) +(i + j − 1)A = cβ i β j ,
ij
(x i+j−1 A ) = c(x i−1 β i )(x j−1 β j ).
ij
In particular,
2
11
(A ) = −α ,
1
2
(xA ) = cβ . (6.3.11)
11
1