Determinants and Their Applications in Mathematical Physics

148 4. Particular Determinants

Theorem 4.52.

(xG

′

n

)

′

=Kn(h)x

h

P

2

n

(x, h),

where

Pn(x, h)=

D

h+n

[x

n

(1 +x)

h+n− 1

]

(h+n−1)!

.

Proof. Referring to (4.10.8),

Gn(x)=

n

∑

i=1

n

∑

j=1

K

(n)

ij

x

h+i+j− 1

h+i+j− 1

=Kn(h)

n

∑

i=1

n

∑

j=1

VniVnjx

h+i+j− 1

(h+i+j−1)

2

. (4.11.57)

Hence,

(xG

′

n

)

′

=Kn(h)x

h

n

∑

i=1

n

∑

j=1

VniVnjx

i+j− 2

=Kn(h)x

h

P

2

n(x, h), (4.11.58)

where

Pn(x, h)=

n

∑

i=1

(−1)

n+i

Vnix

i− 1

=

n

∑

i=1

(h+n+i−1)!x

i− 1

(i−1)! (n−i)! (h+i−1)!

=

n

∑

i− 1

D

h+n

(x

h+n+i− 1

)

(n−i)! (h+i−1)!

, (4.11.59)

(h+n−1)!Pn(x, h)=

n

∑

i=1

(

h+n− 1

h+i− 1

)

D

h+n

(x

h+n+i− 1

)

=D

h+n

[

x

n

n

∑

i=1

(

h+n− 1

h+i− 1

)

x

h+i− 1

]

=D

h+n

[

x

n

h+n− 1

∑

r=h

(

h+n− 1

r

)

x

r

]

=D

h+n

[

x

n

(1 +x)

h+n− 1

−ph+n− 1 (x)

]

, (4.11.60)

wherepr(x) is a polynomial of degreer. The theorem follows.

Let

E(x)=|eij(x)|n− 1 ,