Page 97 - Determinants and Their Applications in Mathematical Physics

82 4. Particular Determinants

Hence,
WW =[α rs ] n ,
where
n
(r−1)(t−1)−(t−1)(s−1)
α rs = ω
t=1
n
(t−1)(r−s)
= ω ,
t=1
α rr = n. (4.4.14)
Put k = r − s, s = r. Then, referring to (4.4.6),

n

α rs = ω (t−1)k (ω = ω = ω k )
k
k
1
t=1
n
t−1
= ω
k
t=1
=0, s = r. (4.4.15)
Hence,
[α rs ]= nI,
WW = nI.
The lemma follows.
The n generalized hyperbolic functions H r ,1 ≤ r ≤ n, of the (n − 1)
independent variables x r ,1 ≤ r ≤ n−1, are defined by the matrix equation
1
H = WE, (4.4.16)
n
where H and E are column vectors defined as follows:

,
T
H = H 1 H 2 H 3 ...H n

,
T
E = E 1 E 2 E 3 ...E n

n−1

E r = exp ω (r−1)t x t , 1 ≤ r ≤ n. (4.4.17)
t=1
Lemma 4.19.
n

E r =1.
r=1
Proof. Referring to (4.4.15),
n−1

n n

E r = exp ω (r−1)t x t
r=1 r=1 t=1