Page 338 - Determinants and Their Applications in Mathematical Physics

A.6 The Generalized Geometric Series and Eulerian Polynomials 323

Rodrigues formula.
1 2 d
P n (x)= D (x − 1) , D = ;
n
n
2 n! dx
n
Generating function relation.
∞
1
2 −
(1 − 2xh + h ) 2 = P n (x)h ;
n
n=0
Recurrence relations.
(n +1)P n+1 (x) − (2n +1)xP n (x)+ nP n−1 (x)=0,
2

(x − 1)P (x)= n[xP n (x) − P n−1 (x)];
n
Differential equation.
2

(1 − x )P (x) − 2xP (x)+ n(n +1)P n (x)=0;
n n
Appell relation. If
2 −n/2
φ n (x)=(1 − x ) P n (x),
then

φ (x)= nFφ n−1 (x),
n
where
2 −3/2
F =(1 − x ) .
A.6 The Generalized Geometric Series and
Eulerian Polynomials

The generalized geometric series φ m (x) and the closely related function
ψ m (x) are defined as follows:
∞

φ m (x)= r x , (A.6.1)
m r
r=0
∞

ψ m (x)= r x . (A.6.2)
m r
r=1
The two sums differ only in their lower limits:
φ m (x)= ψ m (x), m > 0,
1
φ 0 (x)= ,
1 − x
x
ψ 0 (x)=
1 − x
= xφ 0 (x)
= φ 0 (x) − 1. (A.6.3)