For general values of and in Definition 2.1 we haveand 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  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  additional equations relating the quantities , , () and u,v,w.
Consider the kernel K of Definition 2.1 with and defined by (2.18) with m,A,B,C as in (2.19a). We havewhere 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 formulausing (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).