Determinants and Their Applications in Mathematical Physics

6.11 The Relativistic Toda Equation — A Brief Note 303

x=t

√

1 −a

2

=

ct

√

1+c

2

. (6.11.4)

Equations (6.11.1)–(6.11.3) are satisfied by the functions

Un=|ui,n+j− 1 |m,

Vn=|vi,n+j− 1 |m,

Wn=|wi,n+j− 1 |m, (6.11.5)

where the determinants are Casoratians (Section 4.14) of arbitrary order

mwhose elements are given by

uij=Fij+Gij,

vij=aiFij+

1

ai

Gij,

wij=

1

ai

Fij+aiGij, (6.11.6)

where

Fij=

(

1

ai−a

)j

exp(ξi),

Gij=

(

ai

1 −aai

)j

exp(ηi),

ξi=

x

ai

+bi,

ηi=aix+ci, (6.11.7)

and where theai,bi, andciare arbitrary constants.