Determinants and Their Applications in Mathematical Physics

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),