Determinants and Their Applications in Mathematical Physics

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)