In this section, we prove Theorems 2.7 and  2.8.

5.1 Bulk and Edge Scaling Limits for the Mean Diagonal Overlap Proof of Theorem 2.7

By (2.22), for \(a\in \mathbb \), one can write

$$\begin \widehat_^(a) = \bigl (\partial _x\widehat}}}_^(x,a)\bigr |_ \bigr )^ \frac} \lim _ \frac_^(u)}. \end$$

(5.1)

For \(p \in [-1,1]\), we set

$$ z=\sqrtp+\zeta , \qquad u=\sqrtp+\xi ,\qquad a=\sqrtp+\chi , $$

where \(\zeta \in \mathbb \) and \(\xi ,\chi \in \mathbb \).

We first show the bulk case in Theorem 2.7, i.e. \(p\in (-1,1).\) By using (4.24) and [3, Eq.(2.6)], there exists a small positive \(\epsilon >0\) such that for a sufficiently large N,

$$\begin \widehat}_N(z,a) = \widehat}_\textrm(\zeta ,\chi )(1+O(e^)), \end$$

(5.2)

where

$$\begin \widehat}_\textrm(\zeta ,\chi ) = (2(\zeta -\chi )^2-1)\sqrte^\operatorname (\zeta -\chi ) + 2(\zeta -\chi ). \end$$

(5.3)

On the other hand, by (2.9), we have

$$\begin e^f_N(a^2)=N(1-p^2)(1+O(e^)), \end$$

(5.4)

see, e.g. [21, Proposition 2.4]. By combining (5.2) with (5.4) and Stirling’s formula, we have

$$\begin (2N+3)!!\widehat}_(z,a)-2a^2(2N+1)!!\widehat}_(z,a) & =(1-p^2)\widehat}_\textrm(\zeta ,\chi )2^N^\nonumber \\ & \quad \times e^(1+O(N^)). \end$$

(5.5)

Similarly, by combining (4.20) with (5.2), (5.4), (5.5), and Stirling’s formula, we have

$$\begin \widehat_^(z) = (1-p^2) 2^N^e^ \widehat}_\textrm(\zeta ,\chi )(1+O(N^)). \end$$

(5.6)

It also follows from [18, Proposition 3.1] that

$$ \bigl (\partial _x \widehat}}}_^(x,a)\bigr |_ \bigr )^ = \frac+O(e^). $$

Therefore, by (5.1), (5.4), (5.6), Stirling’s formula, and Taylor expansion for (5.3), as \(N\rightarrow \infty \), we have

$$ \bigl (\partial _x\widehat}}}_^(x,a)\bigr |_ \bigr )^e^ \frac^(a)}^} = \fracN(1-p^2)(1+o(1)), $$

uniformly for \(\chi \) in a compact subset of \(\mathbb \). This completes the proof of the bulk case.

Next, we complete the proof of the edge case in Theorem 2.7. Due to the symmetry, it suffices to consider the case \(p=1\). Recall that by [62, Eq.(8.8.9)], the incomplete gamma function satisfies the asymptotic behaviour

$$\begin Q(s+1,s+\sqrtz) = \frac\operatorname \bigl (\frac}\bigr ) + \frac}}} \frac \frac}+O\Bigl (\frac\Bigr ), \qquad s\rightarrow \infty , \end$$

(5.7)

uniformly for z in a compact subset of \(\mathbb \). From [3, Proof of Theorem 2.1] or [18, Proposition 3.1], as \(N\rightarrow \infty \), we have

$$\begin \widehat}}}_N^(z,a) = \widehat_\textrm^(\zeta ,\chi )+O(N^), \end$$

(5.8)

where

$$\begin \begin \widehat_\textrm^(\zeta ,\chi )&= \frac}} \int _^e^\operatorname (\sqrt(\chi -u))\\&\quad -e^\operatorname (\sqrt(\zeta -u)) \, du. \end \end$$

(5.9)

Note here that (5.9) satisfies

$$\begin \partial _\widehat_\textrm^(\zeta ,\chi ) = 2(\zeta -\chi )\widehat_\textrm^(\zeta ,\chi ) + \operatorname (\zeta +\chi )-\frac}}\operatorname (\sqrt\chi ).\quad \end$$

(5.10)

We write

$$\begin (2N+1)\widehat}_(w,a)-2a^2\widehat}_(w,a) = \widehat}_N^(w,a) + \widehat}_N^(w,a), \end$$

(5.11)

where

$$\begin \widehat}_N^(w,a)&:= 2N(\widehat}_(w,a)-\widehat}_(w,a)) -4\chi \sqrt\widehat}_(w,a), \end$$

(5.12)

$$\begin \widehat}_N^(w,a)&:= \widehat}_(w,a)-2\chi ^2\widehat}_(w,a). \end$$

(5.13)

By (2.24) and (5.8), as \(N\rightarrow \infty \), we have

$$ \widehat}_N(w,a) = (\eta -\chi )^2\partial _ \Big [\frac_\textrm^(\eta ,\chi )}\Big ]+O(N^). $$

Furthermore, it follows from (1.17) and (2.24) that

$$\begin \begin&2N(\widehat}_(w,a)-\widehat}_(w,a)) = 2N(w-a)^2\partial _w \frac}}}_^(w,a)-\widehat}}}_^(w,a)} \\ &\quad = \sqrt (\eta -\chi )^2 \partial _ \frac(e^\operatorname (\sqrt\chi )-e^\operatorname (\sqrt\eta ))}(\eta -\chi )}\\ &\qquad \times (1+O(N^)). \end \end$$

(5.14)

Therefore, by (5.11), we have

$$\begin \begin&2^ N^e^(2N1)!!\bigl ( (2N+1)\widehat}_(z,a)-2a^2\widehat}_N(z,a)\bigr ) \\&\quad = (\zeta -\chi )^2\partial _\Bigl [ \frac\bigl (e^\operatorname (\sqrt\chi )-e^\operatorname (\sqrt\zeta )\bigr )-4\sqrt\chi \widehat_\textrm^(\zeta ,\chi )} \Bigr ]\\ &\qquad \times (1+O(N^)), \end \end$$

(5.15)

as \(N\rightarrow \infty \).

On the other hand, by (5.7), we have

$$\begin 2^ae^Q(N+1,a^2)z^e^ & = 2^N^e^e^\nonumber \\ & \quad \times \operatorname (\sqrt\chi ) (1+O(N^)). \end$$

(5.16)

Let us choose \(u=\sqrt+\xi \) for \(\xi \in \mathbb \) so that \(u\rightarrow a\), when \(\xi \rightarrow \chi \). Combining (5.15) and (5.16) with (4.23), we have

$$\begin \begin \widehat_^(u)&\sim 2^ N^e^ \frac\Bigl ( \sqrt}e^ + 2\chi e^\operatorname (\sqrt\chi )\\ &\quad -4\sqrt\chi \operatorname (2\chi ) \Bigr ) (\xi -\chi )^3. \end \end$$

(5.17)

Hence, combining (5.1), (5.10), (5.15), (5.16), (5.17), together with Stirling’s formula, the proof is complete.\(\square \)

5.2 Bulk and Edge Scaling Limits for the Eigenvalue Correlation Functions of the Overlap Weight

In this subsection, we prove Theorem 2.8. According to (2.18) and (2.26), we first define

$$\begin \widetilde}}}_N^(z,w) := e^ (z-a)^4(w-a)^4 }}_N^(z,w). \end$$

(5.18)

Then by (1.9), we have

$$\begin \begin&\quad \textbf_^(z_1,z_2,\dots ,z_k) = \bigg [\begin }}_^(z_j,z_) & }}_^(z_j,\overline_) \\ }}_^(\overline_j,z_) & }}_^(\overline_j,\overline_) \end \bigg ]_^k\\ &\quad \times \prod _^k (\overline_j-z_j) \omega ^)}(z_j) \\&= \bigg [\begin \dfrac}}}_^(z_j,z_)\phi _a(z_)}-\chi ))^4} & \dfrac}}}_^(z_j,\overline_)\phi _a^(z_)}_-\chi ))^4} \\ \dfrac(z_j)\widetilde}}}_^(\overline_j,z_)\phi _a(z_\ell )}_j-\chi )(\zeta _-\chi ))^4} & \dfrac(z_j)\widetilde}}}_^(\overline_j,\overline_)\phi _a^(z_)}_j-\chi )(\overline_-\chi ))^4} \end \bigg ]_^k\\ &\quad \times \prod _^k (\overline_j-\zeta _j) \omega _s(\zeta _j), \end \end$$

(5.19)

where \(\omega _s(\zeta )\) is given by (2.41), and \(\phi _a(z)=e^(z)}\) for \(z\in \mathbb \) and \(a\in \mathbb \). For \(j=1,\dots ,k\), let

$$ z_j=\sqrtp+\zeta _j, \qquad p \in [-1,1]. $$

We also write

$$\begin z=\sqrtp+\zeta , \qquad w=\sqrtp+\eta , \qquad a=\sqrtp+\chi \end$$

(5.20)

for \(\zeta ,\eta \in \mathbb \) and \(\chi \in \mathbb \). We first prove the bulk case of Theorem 2.8.

Proof of Theorem 2.8: bulk case

By (2.19), let us denote

$$\begin \widetilde_\textrm^(\zeta ,\eta ) :=\lim _\widetilde}}}_N^(z,w). \end$$

(5.21)

Then, by Theorem 2.6, (5.2) and Stirling’s formula, we have

$$\begin \Bigl [\partial _\zeta ^2-2\frac\partial _-2\Bigr ] \widetilde_\textrm^(\zeta ,\eta |\chi ) = 4(\zeta -\chi )^2(\eta -\chi )^3e^. \end$$

(5.22)

Since the differential operator is translation invariant under a shift with respect to the horizontal direction along the real line, and the inhomogeneous term in (5.22) is same as (3.9) shifted by \(\chi \), we find that the unique solution for (5.22) with the initial conditions

$$ \widetilde_\textrm^(\zeta ,\zeta )=0, \qquad \partial _\widetilde_\textrm^(\zeta ,\eta )|_=0 $$

is given by

$$\begin \widetilde_\textrm^(\zeta ,\eta ) = (\zeta -\chi )^3 (\eta -\chi )^3\, \varkappa _^)}(\zeta -\chi ,\eta -\chi ). \end$$

(5.23)

Here, \(\varkappa _^)}(z,w)\) is given by (3.8).

For \(z,w\in \mathbb \) and \(u,a\in \mathbb \), according to Proposition 2.4, we define

$$\begin }}_^)}(z,w) & := \frac \bigg ( }}_^)}(z,w) -}}_^)}(z,u)\frac^)}(w)}^)}(u)}\nonumber \\ & \quad +}}_^)}(w,u)\frac^)}(z)}^)}(u)} \bigg ). \end$$

(5.24)

Let us denote

$$\begin \widetilde}}}_^(z,w) := ((z-a)(w-a))^4e^}}_^)}(z,w). \end$$

(5.25)

Clearly, we have

$$\widetilde}}}_^(z,w)\rightarrow \widetilde}}}_^(z,w), \qquad u \rightarrow a. $$

Letting \(u=\sqrtp+\xi \rightarrow a=\sqrtp+\chi \) with \(p\in (-1,1)\) and \(\xi ,\chi \in \mathbb \), it follows from (5.6), (5.21), and (5.23) that as \(N \rightarrow \infty ,\)

$$\begin \widetilde}}}_^(z,w) \sim \widetilde_\textrm^(\zeta ,\eta ) - \widetilde_\textrm^(\zeta ,\xi ) \frac}_\textrm(\eta ,\chi )}}_\textrm(\xi ,\chi )} + \widetilde_\textrm^(\eta ,\xi ) \frac}_\textrm(\zeta ,\chi )}}_\textrm(\xi ,\chi )}, \end$$

(5.26)

uniformly for \(\zeta ,\eta \) in compact subsets of \(\mathbb \) and \(\xi ,\chi \) in a compact subset of \(\mathbb \). Note that by Taylor expansion, as \(\xi \rightarrow \chi \), we have

$$ \frac_\textrm^(\zeta ,\xi )}}_\textrm(\xi ,\chi )} = \frac\bigl ((2(\zeta -\chi )^2-1)e^-(\zeta -\chi )^2+1 \bigr ) +O(\zeta -\chi ). $$

Therefore, combining (5.26) with the above and dividing it by \((\zeta -\chi )^4(\eta -\chi )^4\), we obtain (2.36). This completes the proof of the bulk case.

We now prove the edge case of Theorem 2.8. For this purpose, we need some preparations. First, let us denote

$$\begin F(\chi )&:=e^-\sqrt\chi \operatorname (\sqrt\chi ), \end$$

(5.27)

$$\begin \mathcal _1(\zeta ,\chi )&:=e^(2(\zeta -\chi )^2-1), \end$$

(5.28)

$$\begin \mathcal _2(\zeta ,\chi )&:=\sqrte^\operatorname (\zeta -\chi )(2(\zeta -\chi )^2-1)+2(\zeta -\chi ), \end$$

(5.29)

$$\begin \mathcal (\zeta ,\chi )&:=-8e^(\zeta -\chi )^2, \end$$

(5.30)

$$\begin \mathcal (\eta ,\chi )&:= \partial _\bigg [ \frac(e^\operatorname (\sqrt\chi )-e^\operatorname (\sqrt\eta ))-4\sqrt\chi \widehat_\textrm^(\eta ,\chi )} \bigg ], \end$$

(5.31)

$$\begin \mathcal (\zeta ,\eta |\chi )&:=2(\zeta -\chi )^3(\eta -\chi )^3e^\operatorname (\zeta +\eta ) \nonumber \\ &\quad - \frac}}\nonumber \\ &\quad \times \Bigl (1 -\sqrt}(\zeta +\chi )\frac(\sqrt\chi )}\Bigr )\nonumber \\ &\quad -\frac}\partial _\Big [\frac_\text ^(\eta ,\chi )}\Big ] \nonumber \\ &\quad + \frac}F(\chi )}\mathcal (\eta ,\chi ). \end$$

(5.32)

These are building blocks to define

$$\begin \mathcal (\zeta ,\chi )&:=(\zeta -\chi )^2\Bigl (2e^\operatorname (\sqrt\chi )+\mathcal (\zeta ,\chi )\Bigr ), \end$$

(5.33)

$$\begin \mathcal (\chi )&:=\frac\Bigl [\sqrt}e^+2\chi e^\operatorname (\sqrt\chi )-4\sqrt\chi \operatorname (2\chi ) \Bigr ], \end$$

(5.34)

$$\begin \mathcal (\zeta ,\chi )&:=\frac_2(\zeta ,\chi )} \int _^ \frac_1(\zeta ,\chi )\mathcal _2(t,\chi ) -\mathcal _2(\zeta ,\chi )\mathcal _1(t,\chi )}(t,\chi )}\mathcal (t,\zeta |\chi )\, \hbox t, \end$$

(5.35)

$$\begin \mathcal (\zeta ,\eta |\chi )&:=\frac_1(\zeta ,\chi )\mathcal _2(\eta ,\chi )-\mathcal _1(\eta ,\chi )\mathcal _2(\zeta ,\chi )}_2(\eta ,\chi )} \int _^ \frac_2(t,\chi )\mathcal (t,\eta |\chi )}(t,\chi )}\, \hbox t \nonumber \\&\quad + \int _^ \frac_1(t,\chi )\mathcal _2(\zeta ,\chi )-\mathcal _1(\zeta ,\chi )\mathcal _2(t,\chi )}(t,\chi )} \mathcal (t,\eta |\chi )\,\hbox t. \end$$

(5.36)

Indeed, (5.28) and (5.29) are fundamental solutions to the differential equation (5.42) below, and (5.30) is the Wronskian of (5.28) and (5.29).

Proof of Theorem 2.8: edge case

Let \(z=\sqrt+\zeta ,u=\sqrt+\xi ,a=\sqrt+\chi \) with \(u\rightarrow a\), i.e. as \(\xi \rightarrow \chi \). As before, we first compute the asymptotic behaviour of \(\textrm_N\), \(\textrm_N\), \(\textrm_N\) and \(\textrm_N\) given in (2.28), (2.29), (2.30), and (2.31), respectively. After that, we apply Proposition 2.4.

Recall that \(p=1\). For (2.28) and by using (5.7), as \(N\rightarrow \infty \), we have

$$\begin \textrm_N(z,w) = 2(\zeta -\chi )^3(\eta -\chi )^3e^\operatorname (\zeta +\eta )(1+O(N^)). \end$$

(5.37)

Similarly, for (2.29) and by (5.7), as \(N\rightarrow \infty \), we have

$$\begin \text _N(z,w)=&\frac}}\nonumber \\ &\times \Big ( 1 - \sqrt}(\zeta +\chi ) \frac(\sqrt\chi )} \Big ) (1+O(N^)). \end$$

(5.38)

By (2.30), (2.31) (5.11), (5.12), (5.13), and (5.15), as \(N\rightarrow \infty \), we have

$$\begin&-\textrm_N(z,w)+\textrm_N(z,w) \nonumber \\&\quad \sim -\frac} \partial _ \Big [ \frac_\textrm^(\eta ,\chi )} \Big ] \nonumber \\&\qquad + \frac}F(\chi )} \mathcal (\eta ,\chi ). \end$$

(5.39)

Combining all of the above with (2.28), (2.29), (2.30), and (2.31), as \(N\rightarrow \infty \), we have

$$\begin \textrm_N(z,w) - \textrm_N(z,w) - \textrm_N(z,w) + \textrm_N(z,w) \sim \mathcal (\zeta ,\eta |\chi ), \end$$

(5.40)

uniformly for \(\zeta ,\eta \) in compact subsets of \(\mathbb \) and \(\chi \) in a compact subset of \(\mathbb \), where \(\mathcal (\zeta ,\eta |\chi )\) is given by (5.32).

Now, we are ready to complete the proof. By (5.24) and (5.25), let us denote

$$\begin \widetilde_}^)}(\zeta ,\eta ) = \lim _ \widetilde}}}_N^)}(z,w). \end$$

(5.41)

Then, by Theorem 2.6 and (5.40), we have the following limiting differential equation:

$$\begin \Bigl [\partial _^2-\frac\partial _-2\Bigr ] \widetilde_\textrm^(\zeta ,\eta ) = \frac(\zeta ,\eta |\chi )}. \end$$

(5.42)

As previously mentioned, the fundamental solutions for the homogeneous part of (5.42) are given by (5.28) and (5.29), and their Wronskian is given by (5.30). Therefore, the general solution to (5.42) is given by

$$\begin \widetilde_\textrm^(\zeta ,\eta |\chi ) = c_1\mathcal _1(\zeta ,\chi ) + c_2\mathcal _2(\zeta ,\chi ) + \mathcal (\zeta ,\eta |\chi ), \end$$

(5.43)

where

$$\begin \mathcal (\zeta ,\eta |\chi ) := \int _^ \frac_1(t,\chi )\mathcal _2(\zeta ,\chi )-\mathcal _1(\zeta ,\chi )\mathcal _2(t,\chi )}(t,\chi )} \mathcal (t,\eta |\chi )\, \hbox t. \end$$

(5.44)

We shall determine constants the \(c_1,c_2\), which may depend on \(\eta ,\chi \). Due to skew-symmetry of the kernel, we have

$$\begin c_1\mathcal _1(\eta ,\chi )+c_2\mathcal _2(\eta ,\chi )=0. \end$$

(5.45)

By combining the asymptotic behaviour

$$\begin&\partial _\mathcal _1(\zeta ,\chi )=2(\zeta -\chi )+6(\zeta -\chi )^3+O((\zeta -\chi )^4), \nonumber \\ &\partial _\mathcal _2(\zeta ,\chi )=8(\zeta -\chi )^2+O((\zeta -\chi )^4), \end$$

(5.46)

as \(\zeta \rightarrow \chi \), and

$$\begin \partial _\mathcal (\zeta ,\eta |\chi )&= \int _^ \frac\mathcal _2(\zeta ,\chi )\mathcal _1(t,\chi )}(t,\chi )} \mathcal (t,\eta |\chi ) \, \hbox t \nonumber \\ &\quad - \int _^ \frac\mathcal _1(\zeta ,\chi )\mathcal _2(t,\chi )\mathcal (t,\eta |\chi )}(t,\chi )} \, \hbox t, \end$$

(5.47)

with (5.45), we have

$$ c_1= \int _^ \frac_2(t,\chi )\mathcal (t,\eta |\chi )}(t,\chi )} \, \hbox t,\qquad c_2= -\frac_1(\eta ,\chi )}_2(\eta ,\chi )} \int _^ \frac_2(t,\chi )\mathcal (t,\eta |\chi )}(t,\chi )} \, \hbox t. $$

Therefore, we obtain

$$\begin \widetilde_\textrm^(\zeta ,\eta ) & = \frac_1(\zeta ,\chi )\mathcal _2(\eta ,\chi )-\mathcal _1(\eta ,\chi )\mathcal _2(\zeta ,\chi )}_2(\eta ,\chi )} \nonumber \\ & \quad \times \int _^ \frac_2(t,\chi )\mathcal (t,\eta |\chi )}(t,\chi )}\hbox t + \mathcal (\zeta ,\eta |\chi ) = \mathcal (\zeta ,\eta |\chi ), \end$$

(5.48)

where \(\mathcal (\zeta ,\eta |\chi )\) is given by (5.36). By skew-symmetry of the kernel (5.36), we have

$$\begin \mathcal (\zeta ,u|\chi )&= - \mathcal (u,\zeta |\chi ) \\ &= - \frac_1(u,\chi )\mathcal _2(\zeta ,\chi )-\mathcal _1(\zeta ,\chi )\mathcal _2(u,\chi )}_2(\zeta ,\chi )} \\ &\times \int _^ \frac_2(t,\chi )}(t,\chi )}\mathcal (t,\zeta |\chi )\textt - \mathcal (u,\zeta |\chi ). \end$$

To compute the Taylor expansion of (5.36) at \(u=\chi \), note that

$$\begin \bigl ( \mathcal _1(u,\chi )\mathcal _2(\zeta ,\chi )-\mathcal _1(\zeta ,\chi )\mathcal _2(u,\chi ) \bigr )|_&= -\mathcal _2(\zeta ,\chi ), \\ \partial _\bigl ( \mathcal _1(u,\chi )\mathcal _2(\zeta ,\chi )-\mathcal _1(\zeta ,\chi )\mathcal _2(u,\chi ) \bigr )|_&=0, \\ \partial _^2\bigl ( \mathcal _1(u,\chi )\mathcal _2(\zeta ,\chi )-\mathcal _1(\zeta ,\chi )\mathcal _2(u,\chi ) \bigr )|_&=2\mathcal _2(\zeta ,\chi ), \\ \partial _^3\bigl ( \mathcal _1(u,\chi )\mathcal _2(\zeta ,\chi )-\mathcal _1(\zeta ,\chi )\mathcal _2(u,\chi ) \bigr )|_&=-16\mathcal _1(\zeta ,\chi ), \end$$

and

$$\begin (\mathcal _1(t,\chi )\mathcal _2(u,\chi )-\mathcal _1(u,\chi )\mathcal _2(t,\chi ))|_&=0, \\ \partial _(\mathcal _1(t,\chi )\mathcal _2(u,\chi )-\mathcal _1(u,\chi )\mathcal _2(t,\chi ))|_&=-8e^(u-\chi )^2 , \\ \partial _^2 (\mathcal _1(t,\chi )\mathcal _2(u,\chi )-\mathcal _1(u,\chi )\mathcal _2(t,\chi ))|_&=-16e^(u-\chi )((u-\chi )^2+1). \end$$

By Leibniz integral rule, we have

$$\begin \partial _u^2\mathcal (u,\zeta |\chi )&= \frac(u,\zeta |\chi )} + \int _^u \frac_1(t,\chi )\mathcal _2(u,\chi )-\mathcal _1(u,\chi )\mathcal _2(t,\chi )\bigr )}(t,\chi )} \mathcal (t,\zeta |\chi ) \, \hbox t, \\ \partial _u^3\mathcal (u,\zeta |\chi )&= \partial _u \frac(u,\zeta |\chi )} + 2\frac \mathcal (u,\zeta |\chi ) \\ &\quad + \int _^u \frac_1(t,\chi )\mathcal _2(u,\chi )-\mathcal _1(u,\chi )\mathcal _2(t,\chi )\bigr )}(t,\chi )} \mathcal (t,\zeta |\chi ) \, \hbox t. \end$$

Combining the above with (5.32) and (5.46), we have

$$ \partial _^ \mathcal (\zeta ,u|\chi ) |_ =0,\qquad j=0,1,2, $$

and

$$\begin \partial _^3 \mathcal (\zeta ,u|\chi ) |_&= -\partial _^3 \mathcal (u,\zeta |\chi ) |_\nonumber \\ &= \frac_2(\zeta ,\chi )} \int _^ \frac_1(\zeta ,\chi )\mathcal _2(t,\chi ) -\mathcal _2(\zeta ,\chi )\mathcal _1(t,\chi )}(t,\chi )}\mathcal (t,\zeta |\chi ) \, \hbox t. \end$$

(5.49)

Therefore, as \(u\rightarrow \chi \), we obtain

$$ \widetilde_}}^)}(\zeta ,u) = \mathcal (\zeta ,\chi ) (u-\chi )^3 +O\bigl ((u-\chi )^4 \bigr ), $$

where \(\mathcal (\zeta ,\chi )\) is given by (5.35). By (4.23), (5.15), (5.16), and (5.17), as \(N\rightarrow \infty \), we have

$$\begin \frac_^(z)}_^(u)} \sim \frac(\zeta ,\chi )}(\chi )}, \end$$

(5.50)

uniformly for \(\zeta \) in a compact subset of \(\mathbb \) and \(\chi \) in a compact subset of \(\mathbb \), where \(\mathcal (\zeta ,\chi )\) and \(\mathcal (\chi )\) are given by (5.33) and (5.34). Hence, combining all of the above with (5.24) and (5.25), we have

$$\begin \lim _ \widetilde}}}_^(z,w) = \widetilde_}^(\zeta ,\eta ) \end$$

(5.51)

uniformly for \(\zeta ,\eta \) in compact subsets of \(\mathbb \), and for \(\chi \) in a compact subset of \(\mathbb \), where

$$\begin \begin \widetilde_}^(\zeta ,\eta )&= \mathcal (\zeta ,\eta |\chi ) - \frac(\eta ,\chi )\mathcal (\zeta ,\chi )}(\chi )} + \frac(\zeta ,\chi )\mathcal (\eta ,\chi )}(\chi )}. \end \end$$

(5.52)

By dividing (5.52) by \((\zeta -\chi )^4(\eta -\chi )^4\) and combining it with (5.19), we obtain (2.39). This completes the proof.