Determinants and Their Applications in Mathematical Physics

92 4. Particular Determinants

4.6.2 A ReciprocalPowerSeries

Theorem 4.21. If

∞

∑

r=0

(−1)

r

ψnt

r

=

[

∞

∑

r=0

φrt

r

]− 1

,φ 0 =ψ 0 =1,

then

ψr=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

φ 1 φ 0

φ 2 φ 1 φ 0

φ 3 φ 2 φ 1 φ 0

.....................

φn− 1 φn− 2 ... ... φ 1 φ 0

φn φn− 1 ... ... φ 2 φ 1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

,

which is a Hessenbergian.

Proof. The given equation can be expressed in the form

(φ 0 +φ 1 t+φ 2 t

2

+φ 3 t

3

+···)(ψ 0 −ψ 1 t+ψ 2 t

2

−ψ 3 t

3

+···)=1.

Equating coefficients of powers oft,

n

∑

i=0

(−1)

i+1

φiψn−i= 0 (4.6.4)

from which it follows that

φn=

n

∑

i=1

(−1)

i+1

φn−iψi. (4.6.5)

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

.............................................

φn− 1 ψ 1 −φn− 2 ψ 2 +···+(−1)

n+1

φ 0 ψn=φn.

These arenequations in thenvariables (−1)
r− 1
ψr,1≤r≤n, in which

the determinant of the coefficients is triangular and equal to 1. Hence,

(−1)

n− 1

ψn=

∣

∣

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣

φ 0 φ 1

φ 1 φ 0 φ 2

φ 2 φ 1 φ 0 φ 3

........................................

φn− 2 φn− 3 φn− 4 ··· φ 1 φ 0 φn− 1

φn− 1 φn− 2 φn− 3 ··· φ 2 φ 1 φn

∣

∣

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ n

.

The proof is completed by transferring the last column to the first position,

an operation which introduces the factor (−1)
n− 1

.