Determinants and Their Applications in Mathematical Physics

150 4. Particular Determinants

Note thatUij=Vijin general. Since

ψ

′

m=−mψm−^1 ,

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

i+j+1

∣

∣

∣

∣

n− 1

. (4.11.69)

ExpandUandVas a polynomial inc:

U(x, c)=V(x, c)=

n

∑

r=0

fr(x)c

n−r

. (4.11.70)

However, since

ψm=ymc+zm,

wherezmis independent ofc,

ym=(−1)

m

[

(1 +x)

m+1

−x

m+1

m+1

]

, (4.11.71)

y

′

m

=−mym− 1 ,

y 0 =1, (4.11.72)

it follows from the first line of (4.11.67) thatf 0 , the coefficient ofc

n

inV,

is given by

f 0 =|ym|n

= constant. (4.11.73)

c

n− 1

V(x, c

− 1

)=f 0 c

− 1

+f 1 +

n− 1

∑

r=1

fr+1c

r

,

where

f

′

1

=

[

c

n− 1

DxV(x, c

− 1

)

]

c=0

,Dx=

∂

∂x

,

=

[

c

n− 1

V 11 (x, c

− 1

)

]

c=0

=

[

c

n− 1

∣

∣

∣

∣

c

− 1

(1 +x)

i+j+1

+(1−c

− 1

)x

i+j+1

i+j+1

∣

∣

∣

∣

n− 1

]

c=0

=E. (4.11.74)

Furthermore,

Dc{c

n

U(x, c

− 1

)}=Dc{c

n

V(x, c

− 1

)},Dc=

∂

∂c

,