Page 187 - Determinants and Their Applications in Mathematical Physics
P. 187
172 5. Further Determinant Theory
5.1.2 The Generalized Geometric Series and Eulerian
Polynomials
Notes on the generalized geometric series ψ n (x) and the Eulerian
polynomials A n (x) are given in Appendix A.6.
A n (x)=(1 − x) n+1 ψ n (x). (5.1.2)
Theorem (Lawden).
1 1 − x
1/2! 1 1 − x
1/3! 1/2! 1 1 − x
.
A n
n! x = ....................................
1/(n − 1)! 1/(n − 2)! ··· 1 1 − 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
m
∆ ψ 0 = (−1) m−s ψ s = xψ m . (5.1.3)
m
s
s=0
Put
ψ s =(−1) s! φ s . (5.1.4)
s
Then,
m−1
+(1 − x)φ m =0, m =1, 2, 3,... . (5.1.5)
φ s
(m − s)!
s=0
In some detail,
φ 0 +(1 − x)φ 1 =0,
φ 0 /2! + φ 1 +(1 − x)φ 2 =0,
(5.1.6)
φ 0 /3! + φ 1 /2 + φ 2 +(1 − x)φ 3 =0,
...........................................................
φ 0 /n!+ φ 1 /(n − 1)! + φ 2 /(n − 2)! + ··· + φ n−1 +(1 − x)φ n =0.
When these n equations in the (n + 1) variables φ r ,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),