178 5. Further Determinant Theory
6.Prove that the determinantAnin (5.1.10) satisfies the relation
An+1=vA
′
n
+(u−nv
′
)An.
Putv= 1 to get
An+1=A
′
n
+A 1 An
where
An=
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
uu
′
u
′′
u
′′′
···
− 1 u 2 u
′
3 u
′′
···
− 1 u 3 u
′
···
− 1 u ···
··· ···
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n
These functions appear in a paper by Yebbou on the calculation of
determining factors in the theory of differential equations. Yebbou uses
the notationα
[n]
in place ofAn.
5.2 The Generalized Cusick Identities
The principal Cusick identity in its generalized form relates a particular
skew-symmetric determinant (Section 4.3) to two Hankelians (Section 4.8).
5.2.1 Three Determinants..................
Letφrandψr,r≥1, be two sets of arbitrary functions and define three
power series as follows:
Φi=
∞
∑
r=i
φrt
r−i
,i≥1;
Ψi=
∞
∑
r=i
ψrt
r−i
,i≥1;
Gi=ΦiΨi. (5.2.1)
Let
Gi=
∞
∑
j=i+1
aijt
j−i− 1
,i≥ 1. (5.2.2)
Then, equating coefficients oft
j−i− 1
,
aij=
j−i
∑
s=1
φs+i− 1 ψj−s,i<j. (5.2.3)