, 
. These equations are of two types:
those which apply independent of the particular functions 
and
, and those which are dependent on 
and 
.
For general values of 
and 
in Definition 2.1 we have
and
where j,k = 1,2.
To present the second type of equations, note that the kernel 
as
given by (2.5) is of the type in Definition 2.1 with
In [9] Tracy and Widom show that equations further to those in Proposition
2.3 exist whenever 
and 
satisfy the coupled differential equations
for m,A,B,C polynomials. For the choice (2.6), (2.7) hold with
with
For general 
we can read off from the results of
[9] additional equations relating the quantities 
, 
,
(
) and u,v,w.
Proposition 2.4
Consider the kernel K of Definition 2.1 with 
and 
defined by
(2.18) with m,A,B,C as in (2.19a). We have
where j=1,2.
To pursue our task of deriving the equations in Proposition 2.1, let us return
to the particular case
and 
, 
, 
given by (2.8b) so that 
and 
are given by (2.6). Since
(recall 
) we have 
and thus 
, which together with (2.21) implies
With the aid of (2.22) the equations in Proposition 2.1 can now be deduced in
a straightforward manner from the equations in Proposition 2.3 and 2.4.
We first use (a) and (b) to eliminate 
and 
in (c)-(e) of
Proposition 2.4. Equation (i) of Proposition 2.1 now follows by choosing
j=2 and substituting (2.20) and (2.22). The equations (ii)-(iv) of
Proposition 2.1 are deduced from (c),(d) and (f) of Proposition 2.4
respectively. This requires making use of the general formula
using (2.22), and noting from the first equation in Proposition 2.3 with the
substitutions (2.20) and (2.22) that
The equations (v) follow immediately from the final line of equations in
Proposition 2.3 and (2.22), and the final equation (vi) follows from
the second equation in Proposition 2.3 and (2.23).
![[Annotate]](/organics/icons/sannotate.gif){kind=icon}
![[Shownotes]](../gif/annotate/shide-51.gif)
![[Shownotes]](../gif/annotate/shide-52.gif)