172 5. Further Determinant Theory
5.1.2 The Generalized Geometric Series and Eulerian
Polynomials
Notes on the generalized geometric series ψn(x) and the Eulerian
polynomialsAn(x) are given in Appendix A.6.
An(x)=(1−x)
n+1
ψn(x). (5.1.2)
Theorem (Lawden).
An
n!x
=
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
11 −x
1 /2! 1 1 −x
1 /3! 1 /2! 1 1 −x
....................................
1 /(n−1)! 1/(n−2)! ··· 11 −x
1 /n!1/(n−1)! ··· 1 /2! 1
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n
.
The determinant is a Hessenbergian.
Proof. It is proved in the section on differences (Appendix A.8) that
∆
m
ψ 0 =
m
∑
s=0
(−1)
m−s
(
m
s
)
ψs=xψm. (5.1.3)
Put
ψs=(−1)
s
s!φs. (5.1.4)
Then,
m− 1
∑
s=0
φs
(m−s)!
+(1−x)φm=0,m=1, 2 , 3 ,.... (5.1.5)
In some detail,
φ 0 +(1−x)φ 1 =0,
φ 0 /2! +φ 1 +(1−x)φ 2 =0,
φ 0 /3! +φ 1 /2+φ 2 +(1−x)φ 3 =0,
...........................................................
φ 0 /n!+φ 1 /(n−1)! +φ 2 /(n−2)! +···+φn− 1 +(1−x)φn=0.
(5.1.6)
When thesenequations in the (n+ 1) variables φr,0≤ r ≤ n, are
augmented by the relation
(1−x)φ 0 =x, (5.1.7)
the determinant of the coefficients is triangular so that its value is
(1−x)
n+1
. Solving the (n+ 1) equations by Cramer’s formula (Sec-
tion 2.3.5),