Determinants and Their Applications in Mathematical Physics

4.11 Hankelians 4 149

where

eij(x)=

(1 +x)

i+j+1

−x

i+j+1

i+j+1

. (4.11.61)

Theorem 4.53. The polynomial determinantE satisfies the nonlinear

differential equation

[

{x(1 +x)E}

′′

] 2

=4n

2

(xE)

′

{(1 +x)E}

′

.

Proof. Let

A(x, ξ)=|φm(x, ξ)|n, 0 ≤m≤ 2 n− 2 ,

where

φm(x, ξ)=

1

m+1

[

(ξ+x)

m+1

−c(ξ−1)

m+1

+(c−1)ξ

m+1

]

. (4.11.62)

Then,

∂

∂ξ

φm(x, ξ)=mφm− 1 (x, ξ),

φ 0 (x, ξ)=x+c. (4.11.63)

Hence, from Theorem 4.33 in Section 4.9.1,Ais independent ofξ. Put

ξ= 0 and−xin turn and denote the resulting determinants byUandV,

respectively. Then,

A=U=V, (4.11.64)

where

U(x, c)=|φm(x,0)|n

=

∣

∣

∣

∣

x

m+1

+(−1)

m

c

m+1

∣

∣

∣

∣

n

, 0 ≤m≤ 2 n− 2

=

∣

∣

∣

∣

x

i+j− 1

+(−1)

i+j

c

i+j− 1

∣

∣

∣

∣

n

. (4.11.65)

Put

ψm(x)=φm(x,−x)

=

(−1)

m

m+1

[c(1 +x)

m+1

+(1−c)x

m+1

] (4.11.66)

V(x, c)=|ψm(x)|n,

=

∣

∣

∣

∣

c(1 +x)

m+1

+(1−c)x

m+1

m+1

∣

∣

∣

∣

n

=

∣

∣

∣

∣

c(1 +x)

i+j− 1

+(1−c)x

i+j− 1

i+j− 1

∣

∣

∣

∣

n

. (4.11.67)