Determinants and Their Applications in Mathematical Physics

4.12 Hankelians 5 157

=2

n− 1

n

∏

i=2

(uv)

i− 1

=2

n− 1

(uv)

1+2+3+···+n− 1

=2

−(n−1)

2

(x

2

−1)

1

2

n(n−1)

.

which completes the proof.

Exercises

1.Prove that

|Hm(x)|n=(−2)

n(n−1)/ 2

1! 2! 3!···(n−1)!,

0 ≤m≤ 2 n− 2

whereHm(x) is the Hermite polynomial.

2.If

An=

∣

∣

∣

∣

Pn− 1 Pn

Pn Pn+1

∣

∣

∣

∣

,

prove that

n(n+1)A

′′

n=2(P

′

n)

2

. (Beckenbach et al.)

4.12.2 The Generalized Geometric Series and Eulerian

Polynomials

Notes on the generalized geometric seriesφm(x), the closely related function

ψm(x), the Eulerian polynomialAn(x), and Lawden’s polynomialSn(x) are

given in Appendix A.6.

ψm(x)=

∞

∑

r=1

r

m

x

r

,

xψ

′

m(x)=ψm+1(x), (4.12.13)

Sm(x)=(1−x)

m+1

ψm,m≥ 0 , (4.12.14)

Am(x)=Sm(x),m> 0 ,

A 0 =1,S 0 =x. (4.12.15)

Theorem (Lawden).

En=|ψi+j− 2 |n=

λnx

n(n+1)/ 2

(1−x)

n^2

,

Fn=|ψi+j− 1 |n=

λnn!x

n(n+1)/ 2

(1−x)

n(n+1)

,

Gn=|ψi+j|n=

λn(n!)

2

x

n(n+1)/ 2

(1−x

n+1

)

(1−x)

(n+1)^2

,