Determinants and Their Applications in Mathematical Physics

A.6 The Generalized Geometric Series and Eulerian Polynomials 323

Rodrigues formula.

Pn(x)=

1

2

n

n!

D

n

(x

2

−1)

n

,D=

d

dx

;

Generating function relation.

(1− 2 xh+h

2

)

−

1

(^2) =
∞
∑
n=0
Pn(x)h
n
;
Recurrence relations.
(n+1)Pn+1(x)−(2n+1)xPn(x)+nPn− 1 (x)=0,
(x
2
−1)P
′
n(x)=n[xPn(x)−Pn−^1 (x)];
Differential equation.
(1−x
2
)P
′′
n
(x)− 2 xP
′
n
(x)+n(n+1)Pn(x)=0;
Appell relation.If
φn(x)=(1−x
2
)
−n/ 2
Pn(x),
then
φ
′
n
(x)=nF φn− 1 (x),
where
F=(1−x
2
)
− 3 / 2
.

A.6 The Generalized Geometric Series and

Eulerian Polynomials

The generalized geometric seriesφm(x) and the closely related function

ψm(x) are defined as follows:

φm(x)=

∞

∑

r=0

r

m

x

r

, (A.6.1)

ψm(x)=

∞

∑

r=1

r

m

x

r

. (A.6.2)

The two sums differ only in their lower limits:

φm(x)=ψm(x),m> 0 ,

φ 0 (x)=

1

1 −x

,

ψ 0 (x)=

x

1 −x

=xφ 0 (x)

=φ 0 (x)− 1. (A.6.3)