Determinants and Their Applications in Mathematical Physics

4.11 Hankelians 4 151

=Dc

n

∑

r=0

frc

r

=f 1 +

n

∑

r=2

rfrc

r− 1

. (4.11.75)

Hence,

f 1 =

[

Dc{c

n

U(x, c

− 1

)}

]

c=0

=Dc

[

c

n

∣

∣

∣

∣

x

i+j− 1

−(−1)

i+j

c

− 1

i+j− 1

∣

∣

∣

∣

n

]

c=0

=

[

Dc

∣

∣

∣

∣

cx

i+j− 1

−(−1)

i+j

i+j− 1

∣

∣

∣

∣

n

]

c=0

=

n

∑

k=1

Gn(x, 0 ,k)

=Gn(x,0), (4.11.76)

whereGn(x, h, k) andGn(x, h) are defined in the first line of (4.11.55) and

(4.11.56), respectively.

E=G

′

,

(xE)

′

=(xG

′

)

′

=KnP

2

n

, (4.11.77)

where

Kn=Kn(0),

Pn=Pn(x,0)

=

D

n

[x

n

(1 +x)

n− 1

]

(n−1)!

. (4.11.78)

Let

Qn=

D

n

[x

n− 1

(1 +x)

n

]

(n−1)!

. (4.11.79)

Then,

Pn(− 1 −x)=(−1)

n

Qn.

Since

E(− 1 −x)=E(x),

it follows that

{(1 +x)E}

′

=KnQ

2

n

,

{xE}

′

{(1 +x)E}

′

=(KnPnQn)

2

. (4.11.80)