Determinants and Their Applications in Mathematical Physics

82 4. Particular Determinants

Hence,

WW=[αrs]n,

where

αrs=

n

∑

t=1

ω

(r−1)(t−1)−(t−1)(s−1)

=

n

∑

t=1

ω

(t−1)(r−s)

,

αrr=n. (4.4.14)

Putk=r−s,s=r. Then, referring to (4.4.6),

αrs=

n

∑

t=1

ω

(t−1)k

(ω

k

=ω

k

1

=ωk)

=

n

∑

t=1

ω

t− 1

k

=0,s=r. (4.4.15)

Hence,

[αrs]=nI,

WW=nI.

The lemma follows.

Thengeneralized hyperbolic functionsHr,1≤r≤n, of the (n−1)

independent variablesxr,1≤r≤n−1, are defined by the matrix equation

H=

1

n

WE, (4.4.16)

whereHandEare column vectors defined as follows:

H=

[

H 1 H 2 H 3 ...Hn

]T

,

E=

[

E 1 E 2 E 3 ...En

]

T

,

Er= exp

[

n− 1

∑

t=1

ω

(r−1)t

xt

]

, 1 ≤r≤n. (4.4.17)

Lemma 4.19.

n

∏

r=1

Er=1.

Proof. Referring to (4.4.15),

n

∏

r=1

Er=

n

∏

r=1

exp

[

n− 1

∑

t=1

ω

(r−1)t

xt

]