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To derive the equations in Proposition 2.1 we use the theory of [9] to first
obtain some equations for the quantities in Definition 2.1 considered as
functions of the end points , . These equations are of two types:
those which apply independent of the particular functions and
, and those which are dependent on and .

The equations of interest which fall into the first category are summarized
as follows.

** Proposition 2.3**
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).

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