Determinants and Their Applications in Mathematical Physics

6.3 The Dale Equation 247

Note that the theorem cannot be applied toAdirectly sinceφmdoes not

satisfy the Appell equation for anyF(x).

Using the identity

|t

i+j− 2

aij|n=t

n(n−1)

|aij|n,

it is found that

P=(1+x)

−n

2

A,

P 11 =(1+x)

−n

2

+1

A 11. (6.3.4)

Hence,

(1 +x)A

′

=n

2

A−(c−1)A 11. (6.3.5)

Let

αi=

∑

r

x

r− 1

A

ri

, (6.3.6)

βi=

∑

r

(−1)

r

A

ri

, (6.3.7)

λ=

∑

r

(−1)

r

αr

=

∑

r

∑

s

(−1)

r

x

s− 1

A

rs

=

∑

s

x

s− 1

βs, (6.3.8)

whererands=1, 2 , 3 ,...,nin all sums.

Applying double-sum identity (D) in Section 3.4 withfr=randgs=

s−1, then (B),

(i+j−1)A

ij

=

∑

r

∑

s

[x

r+s− 1

+(−1)

r+s

c]A

ri

A

sj

=xαiαj+cβiβj (6.3.9)

(A

ij

)

′

=−

∑

r

∑

s

x

i+j− 2

A

is

A

rj

=−αiαj. (6.3.10)

Hence,

x(A

ij

)

′

+(i+j−1)A

ij

=cβiβj,

(x

i+j− 1

A

ij

)

′

=c(x

i− 1

βi)(x

j− 1

βj).

In particular,

(A

11

)

′

=−α

2

1 ,

(xA

11

)

′

=cβ

2

1

. (6.3.11)