Page 193 - Determinants and Their Applications in Mathematical Physics

178 5. Further Determinant Theory

6. Prove that the determinant A n in (5.1.10) satisfies the relation

A n+1 = vA +(u − nv )A n .
n
Put v = 1 to get

A n+1 = A + A 1 A n
n
where
u u u u ···

−1 u 2u 3u ···

−1 u 3u ··· .

A n =
−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 notation α [n] in place of A n .
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
Let φ r and ψ r , r ≥ 1, be two sets of arbitrary functions and define three
power series as follows:
∞

Φ i = φ r t r−i , i ≥ 1;
r=i
∞

Ψ i = ψ r t r−i , i ≥ 1;
r=i
G i =Φ i Ψ i . (5.2.1)
Let
∞
j−i−1
G i = a ij t , i ≥ 1. (5.2.2)
j=i+1
Then, equating coefficients of t j−i−1 ,

j−i

a ij = φ s+i−1 ψ j−s , i < j. (5.2.3)
s=1