Page 107 - Determinants and Their Applications in Mathematical Physics

92 4. Particular Determinants

4.6.2 A Reciprocal Power Series
Theorem 4.21. If
−1

∞ ∞

(−1) ψ n t = φ r t r , φ 0 = ψ 0 =1,
r
r
r=0 r=0
then

φ 1 φ 0

φ 2 φ 1 φ 0

φ 3 φ 2 φ 1 φ 0 ,

.....................
ψ r =

φ n−1 φ n−2 ... ... φ 1 φ 0

φ n−1 ... ... φ 2 φ 1 n
φ n
which is a Hessenbergian.
Proof. The given equation can be expressed in the form
2
2
3
3
(φ 0 + φ 1 t + φ 2 t + φ 3 t + ···)(ψ 0 − ψ 1 t + ψ 2 t − ψ 3 t + ···)=1.
Equating coefficients of powers of t,
n
i+1
(−1) φ i ψ n−i = 0 (4.6.4)
i=0
from which it follows that
n
i+1
φ n = (−1) φ n−i ψ i . (4.6.5)
i=1
In some detail,
φ 0 ψ 1 = φ 1
φ 1 ψ 1 − φ 0 ψ 2 = φ 2
φ 2 ψ 1 − φ 1 ψ 2 + φ 0 ψ 3 = φ 3
.............................................
φ n−1 ψ 1 − φ n−2 ψ 2 + ··· +(−1) n+1 φ 0 ψ n = φ n .
These are n equations in the n variables (−1) r−1 ψ r ,1 ≤ r ≤ n, in which
the determinant of the coefficients is triangular and equal to 1. Hence,

φ 0 φ 1

φ 1 φ 0 φ 2

(−1) n−1 ψ n = φ 2 φ 1 φ 0 φ 3 .
........................................

φ n−2 φ n−3 φ n−4 ··· φ 1 φ 0 φ n−1

φ n−1 φ n−2 φ n−3 ··· φ 2 φ 1 φ n
n
The proof is completed by transferring the last column to the first position,
an operation which introduces the factor (−1) n−1 .