Toeplitz determinants with merging singularities

Theorem 1.1. Let $\alpha_1, \alpha_2, \beta_1, \beta_2 \in \mathbb{C}$ be such that $$\operatorname{Re} \alpha_1, \operatorname{Re} \alpha_2 > -\frac{1}{2}, \qquad \operatorname{Re} (\alpha_1 + \alpha_2) > -\frac{1}{2}, \qquad |||\beta||| < 1,$$ (1.17) and assume that $$\alpha_1 \pm \beta_1, \quad \alpha_2 \pm \beta_2, \quad \alpha_1 + \alpha_2 \pm \beta_1 \pm \beta_2 \notin \{-1, -2, -3, \ldots\}.$$ (1.18) If $2(\alpha_1 + \alpha_2) \notin \mathbb{N} \cup \{0\}$ , there exists a solution $\sigma(s)$ of equation (1.14) with the following asymptotic behavior as $|s| \to 0$ along the negative imaginary axis: $$\sigma(s) = 2\alpha_1 \alpha_2 - \frac{1}{2} (\beta_1 + \beta_2)^2 - \frac{(\alpha_1 - \alpha_2)(\beta_1 + \beta_2)}{2(\alpha_1 + \alpha_2)} s + \tau_0 |s|^{1 + 2(\alpha_1 + \alpha_2)} + \mathcal{O}(|s|^2) + \mathcal{O}(|s|^{2 + 4(\alpha_1 + \alpha_2)}), \qquad s \to -i0_+, \quad (1.19)$$ where $$\tau_{0} = -\frac{\Gamma(1 + \alpha_{1} + \alpha_{2} + \beta_{1} + \beta_{2})\Gamma(1 + \alpha_{1} + \alpha_{2} - \beta_{1} - \beta_{2})}{2\pi\Gamma(1 + 2(\alpha_{1} + \alpha_{2}))^{2}} \frac{\Gamma(1 + 2\alpha_{1})\Gamma(1 + 2\alpha_{2})}{\Gamma(2 + 2(\alpha_{1} + \alpha_{2}))} \times \left[ e^{i\pi(\alpha_{1} - \alpha_{2})} \frac{\sin \pi(\alpha_{1} + \alpha_{2} + \beta_{1} + \beta_{2})}{\sin 2\pi(\alpha_{1} + \alpha_{2})} + e^{-i\pi(\alpha_{1} - \alpha_{2})} \frac{\sin \pi(\alpha_{1} + \alpha_{2} - \beta_{1} - \beta_{2})}{\sin 2\pi(\alpha_{1} + \alpha_{2})} - e^{i\pi(\beta_{1} - \beta_{2})} \right],$$ (1.20) and with the following asymptotic behavior as $|s| \to \infty$ along the negative imaginary axis: $$\sigma(s) = \frac{\beta_2 - \beta_1}{2} s - \frac{1}{2} (\beta_1 - \beta_2)^2 \pm \frac{s\gamma(s)}{1 + \gamma(s)} + \mathcal{O}(|s|^{-1 + ||\beta||}), \qquad s \to -i\infty, (1.21)$$ where "+" is taken for $\operatorname{Re}(\beta_1 - \beta_2) \geq 0$ , "-" for $\operatorname{Re}(\beta_1 - \beta_2) < 0$ , and $$\gamma(s) = \begin{cases} |s|^{2(-1+\beta_1-\beta_2)} e^{-i|s|} e^{i\pi(\alpha_1+\alpha_2)} \frac{\Gamma(1+\alpha_1-\beta_1)\Gamma(1+\alpha_2+\beta_2)}{\Gamma(\alpha_1+\beta_1)\Gamma(\alpha_2-\beta_2)}, & \operatorname{Re}(\beta_1-\beta_2) \ge 0, \\ |s|^{2(-1+\beta_2-\beta_1)} e^{i|s|} e^{-i\pi(\alpha_1+\alpha_2)} \frac{\Gamma(1+\alpha_2-\beta_2)\Gamma(1+\alpha_1+\beta_1)}{\Gamma(\alpha_2+\beta_2)\Gamma(\alpha_1-\beta_1)}, & \operatorname{Re}(\beta_1-\beta_2) < 0. \end{cases}$$ (1.22) If $2(\alpha_1 + \alpha_2) \in \mathbb{N} \cup \{0\}$ , there exists a solution $\sigma(s)$ of equation (1.14) satisfying $$\sigma(s) = 2\alpha_1 \alpha_2 - \frac{1}{2}(\beta_1 + \beta_2)^2 + \mathcal{O}(|s \ln |s||), \qquad s \to -i0_+,$$ (1.23) and satisfying (1.21) as $s \to -i\infty$ . Moreover, if $\alpha_1, \alpha_2 \in \mathbb{R}$ and $\beta_1, \beta_2 \in i\mathbb{R}$ , there exists a solution $\sigma(s)$ which is real and free of poles for $s \in -i\mathbb{R}^+$ , and which has the asymptotics (1.19) (or (1.23) if $2(\alpha_1 + \alpha_2) \in \mathbb{N} \cup \{0\}$ ) and (1.21). Remark 1.2 We will construct solutions $\sigma(s)$ satisfying the above properties in terms of a Riemann-Hilbert (RH) problem which depends on the parameters $\alpha_1, \alpha_2, \beta_1, \beta_2$ and on s. The solutions constructed in this way will be the ones appearing in the asymptotic expansion for the Toeplitz determinants $D_n(f_t)$ . However, we do not prove that there is only one solution $\sigma$ which satisfies the properties given in Theorem 1.1. Remark 1.3 Equation (1.14) depends, through $\theta_1, \ldots, \theta_4$ , on the three independent parameters $\alpha_1, \alpha_2$ , and $\beta_1 + \beta_2$ . On the other hand, the solutions described in the above theorem depend not only on the sum $\beta_1 + \beta_2$ , but also on $\beta_1$ and $\beta_2$ independently. This means that, given $\alpha_1, \alpha_2$ , and $\beta_1 + \beta_2$ , the asymptotics (1.21) and (1.19), (1.23) specify a one-parameter family of solutions to the same differential equation (1.14). Remark 1.4 The function $\sigma(s)$ has a branching point at zero (any other singularities of $\sigma(s)$ are poles) and is defined on the plane with a cut from zero to infinity. The assumption in the theorem that s is on the negative imaginary axis is not essential: it is adopted for simplicity and in view of the application in Theorem 1.5 below. A simple modification of the proof shows that the asymptotics (1.19), (1.21), (1.23) hold along a path from zero to infinity in a neighborhood of the negative imaginary axis. This fact is used in Theorem 1.8 below. We now state the result about Toeplitz determinants for the case $\alpha_j, i\beta_j \in \mathbb{R}$ .

Theorem 1.5. Let $\alpha_1, \alpha_2, \alpha_1 + \alpha_2 > -1/2$ and $\beta_1, \beta_2 \in i\mathbb{R}$ . Let $D_n(f_t)$ be the Toeplitz determinant (1.6) corresponding to the symbol (1.2). The following asymptotic expansion holds as $n \to \infty$ with the error term uniform for $t \in (0, t_0)$ , where $t_0$ is sufficiently small: $$\ln D_n(f_t) = \ln D_n(f_0) + int(\beta_2 - \beta_1) + \int_0^{-2int} \frac{1}{s} \left( \sigma(s) - 2\alpha_1 \alpha_2 + \frac{1}{2} (\beta_1 + \beta_2)^2 \right) ds$$ $$+ 2 \left( \beta_1 \beta_2 - \alpha_1 \alpha_2 \right) \ln \frac{\sin t}{t} + 2it(\alpha_2 \beta_1 - \alpha_1 \beta_2) - \alpha_1 (V(e^{it}) - V(1))$$ $$- \alpha_2 (V(e^{-it}) - V(1)) + \beta_1 \ln \frac{b_+(e^{it})b_-(1)}{b_-(e^{it})b_+(1)} + \beta_2 \ln \frac{b_+(e^{-it})b_-(1)}{b_-(e^{-it})b_+(1)} + o(1), \quad (1.24)$$ where the function $\sigma(s)$ satisfies the conditions of Theorem 1.1: it solves equation (1.14), has the asymptotics (1.19) if $2(\alpha_1 + \alpha_2) \notin \mathbb{N} \cup \{0\}$ ((1.23) otherwise) and (1.21), and has no poles for $s \in -i\mathbb{R}^+$ . Here $\ln D_n(f_0)$ is given by (1.13). Remark 1.6 The integral in (1.24) is well-defined by (1.19), (1.23), and by the fact that $\sigma$ has no poles on the interval of integration. Remark 1.7 If $\alpha_1 = \alpha_2 = \beta_1 = \beta_2 = \frac{1}{2}$ , the function $\sigma(s)$ is identically zero, as we show in Section 3.2. Note that in this case, the parameters $\theta_1, \ldots, \theta_4$ in the Painlevé equation (1.14) are given by $\theta_1 = \theta_3 = 0$ , $\theta_2 = 1$ , $\theta_4 = -1$ , and it is easily verified that $\sigma(s) = 0$ solves (1.14), and that it satisfies the asymptotic conditions (1.23) and (1.21). Although $\beta_1, \beta_2 \notin i\mathbb{R}$ in this case, the asymptotic expansion (1.24) holds and becomes elementary. An extension of the previous theorem to the generic case $|||\beta||| < 1$ is the following.

Theorem 1.8. Let Re $\alpha_1$ , Re $\alpha_2$ , Re $(\alpha_1 + \alpha_2) > -1/2$ , $|||\beta||| < 1$ , and $$\alpha_1 \pm \beta_1, \alpha_2 \pm \beta_2, \alpha_1 + \alpha_2 \pm \beta_1 \pm \beta_2 \notin \{-1, -2, -3, \ldots\}.$$ Let $D_n(f_t)$ be the Toeplitz determinant (1.6) corresponding to the symbol (1.2). There exists a finite set $\Omega = \{s_1, \ldots, s_\ell\} \subset -i\mathbb{R}^+ = (0, -i\infty)$ (with $\ell = \ell(\alpha_1, \alpha_2, \beta_1, \beta_2)$ ) such that the asymptotic expansion (1.24) holds uniformly for $t \in (0, t_0)$ , where $t_0$ is sufficiently small, provided -2int is bounded away from $\Omega$ . The path of integration in the integral on the r.h.s. of (1.24) is chosen in the complex s-plane to avoid the points of $\Omega$ . The function $\sigma(s)$ is a solution to (1.14) with the asymptotics (1.19) (or (1.23) if $2(\alpha_1 + \alpha_2) \in \mathbb{N} \cup \{0\}$ ) and (1.21). Remark 1.9 The set $\Omega$ is the set of points where the Riemann-Hilbert problem associated to $\sigma(s)$ is not solvable. $\Omega$ contains the poles of $\sigma(s)$ . A pole $s_j$ corresponds to a zero in the asymptotics of the determinant $D_n(f_t)$ for $t_j = i s_j/(2n)$ . Different choices of the integration contour in (1.24) correspond to different branches of $\ln D_n$ . For $\alpha_j, i\beta_j \in \mathbb{R}$ , we show in Section 3.4 that $\Omega$ has no points on the half-line $-i\mathbb{R}^+$ , and hence a simpler formulation of the result in Theorem 1.5. Remark 1.10 An estimate for the error term in (1.24) for both theorems is given in the Proposition 8.1 below. If $t \to 0$ sufficiently fast so that also $nt \to 0$ , we immediately obtain (1.13) from (1.24). Let us check that we also recover (1.8) from (1.24) when t is fixed, and so $nt \to \infty$ . Note first that it follows from the asymptotics for $\sigma$ that, given (1.17), (1.18), $$\int_{0}^{-2int} \frac{1}{s} \left( \sigma(s) - 2\alpha_{1}\alpha_{2} + \frac{1}{2}(\beta_{1} + \beta_{2})^{2} \right) ds$$ $$= -int(\beta_{2} - \beta_{1}) - \left( \frac{1}{2}(\beta_{1} - \beta_{2})^{2} + \sigma(0) \right) \ln(2nt) + \mathcal{O}(1)$$ $$= -int(\beta_{2} - \beta_{1}) - 2(\alpha_{1}\alpha_{2} - \beta_{1}\beta_{2}) \ln(2nt) + \mathcal{O}(1), \quad nt \to \infty. \quad (1.25)$$ Substituting this expression into the right hand side of (1.24), we obtain the terms with n and with $\ln n$ in (1.8). Equality of the constant in n terms in both expressions for t fixed gives the following integral identity for $\sigma(s)$ : $$\lim_{T \to +\infty} \left( \int_0^{-iT} \frac{1}{s} \left( \sigma(s) - 2\alpha_1 \alpha_2 + \frac{1}{2} (\beta_1 + \beta_2)^2 \right) ds + \frac{iT}{2} (\beta_2 - \beta_1) + 2(\alpha_1 \alpha_2 - \beta_1 \beta_2) \ln T \right)$$ $$= i\pi (\alpha_1 \beta_2 - \alpha_2 \beta_1) - \ln \frac{G(1 + \alpha_1 + \alpha_1 + \beta_1 + \beta_2) G(1 + \alpha_1 + \alpha_1 - \beta_1 - \beta_2)}{G(1 + 2\alpha_1 + 2\alpha_2)}$$ $$+ \ln \frac{G(1 + \alpha_1 + \beta_1) G(1 + \alpha_1 - \beta_1) G(1 + \alpha_2 + \beta_2) G(1 + \alpha_2 - \beta_2)}{G(1 + 2\alpha_1) G(1 + 2\alpha_2)}. \quad (1.26)$$ This identity is a deep result which contains global information about $\sigma$ . We believe that it is of independent interest in the study of Painlevé transcendents. The following result extends the expansion (1.8), known for fixed singularities $z_1$ , $z_2$ independent of n, to the case where the two singularities approach each other at a sufficiently slow rate as $n \to \infty$ .

Theorem 1.11. Let Re $\alpha_1$ , Re $\alpha_2 > -1/2$ , $|||\beta||| < 1$ , and $\alpha_1 \pm \beta_1$ , $\alpha_2 \pm \beta_2 \notin \{-1, -2, ...\}$ . Let $D_n(f_t)$ be the Toeplitz determinant (1.6) corresponding to the symbol (1.2); $\omega(x)$ be any positive, smooth for large x function such that $\omega(n) \to \infty$ , $\omega(n)/n \to 0$ as $n \to \infty$ . Then the expansion (1.8) holds as $n \to \infty$ for $\omega(n)/n \le t < t_0$ with $t_0$ sufficiently small. The error term in (1.8) in this case is $o(1) = \mathcal{O}(\omega(n)^{-1+|||\beta|||})$ , uniformly in t. Moreover, there exist positive constants $n_0$ , $s_0$ , $c_0$ , depending only on $\alpha_j$ , $\beta_j$ , j = 1, 2, and V(z), such that the expansion (1.8) holds for $n > n_0$ with the error term o(1) replaced by a function v(s), s = nt, satisfying the estimate $|v(s)| < c_0 s^{-1+||\beta|||}$ for $s \ge s_0$ , $t < t_0$ . To complete the analysis of the nondegenerate (by which we mean that the condition (1.18) holds) situation it remains to consider the case $|||\beta||| = 1$ . We have

Theorem 1.12. Let Re $\alpha_1$ , Re $\alpha_2$ , Re $(\alpha_1+\alpha_2) > -1/2$ , and assume (1.18). Let $|||\beta||| = 0$ , and denote $\beta_1^- = \beta_1$ , $\beta_2^- = \beta_2 - 1$ , $f_t^- = f_t(z; \alpha_1, \alpha_2, \beta_1^-, \beta_2^-)$ , $f_t = f_t(z; \alpha_1, \alpha_2, \beta_1, \beta_2)$ . Then $|||\beta^-||| = 1$ . There exists a sufficiently large $C_0$ such that the following asymptotic expansion holds outside the set $\Omega$ of Theorem 1.8: $$D_{n-1}(f_{t}^{-}) = e^{-i(n-1)t}b_{0}^{-1}D_{n}(f_{t})$$ $$\begin{cases} -r(-2int)\frac{b_{-}(1)}{b_{+}(1)}t\left(\frac{nt}{\sin t}\right)^{2(\beta_{1}+\beta_{2})}e^{i\pi(-\alpha_{1}+3\beta_{1}+\alpha_{2}+\beta_{2})}(1+\mathcal{O}(t)), & 0 < t \leq C_{0}/n \\ n^{2\beta_{1}-1}z_{1}^{-n+1}\frac{b_{-}(z_{1})}{b_{+}(z_{1})}\frac{\Gamma(1+\alpha_{1}-\beta_{1})}{\Gamma(\alpha_{1}+\beta_{1})}e^{i(\pi-2t)\alpha_{2}}(2\sin t)^{-2\beta_{2}} \\ \times (1+\mathcal{O}((nt)^{-1})) \\ +n^{2\beta_{2}-1}z_{2}^{-n+1}\frac{b_{-}(z_{2})}{b_{+}(z_{2})}\frac{\Gamma(1+\alpha_{2}-\beta_{2})}{\Gamma(\alpha_{2}+\beta_{2})}e^{i(-\pi+2t)\alpha_{1}}(2\sin t)^{-2\beta_{1}} \\ \times (1+\mathcal{O}((nt)^{-1})), & C_{0}/n < t < t_{0}, \\ (1.27) \end{cases}$$ as $n \to \infty$ , with the error term uniform for -2int bounded away from $\Omega$ . Here the asymptotics for $D_n(f_t)$ are given by Theorem 1.8, and r(s) is a Painlevé V function defined in Section 3.3. In particular, r(s) is related to $\sigma(s)$ by (3.55), and has the large-s asymptotics (9.16) and the small-s asymptotics (9.18). Remark 1.13 The large-s expansion for r(s) implies, by Remark 9.2 below, that the 2 parts of the asymptotics (1.27) coincide in a neighborhood of the boundary $t = C_0/n$ . Thus (1.27) is a complete analogue of (1.24). Remark 1.14 If $\alpha_1 = \alpha_2 = \beta_1 = \beta_2 = \frac{1}{2}$ , the function r(s) is elementary, namely (as we obtain in Section 9.2), $$r(s) = -\frac{\sin nt}{(nt)^2}, \qquad s = -2int.$$ In Section 9.2 we then show that, in this case, the part of (1.27) for $C_0/n < t < t_0$ holds uniformly for the whole range $0 < t < t_0$ . This fact was used in [9] to analyze the eigenvalues of the Toeplitz matrix $T_n(g)$ with a smooth real-valued symbol g in a small neighbourhood of the edge of the spectrum. The rest of the eigenvalues of $T_n(g)$ were analyzed in [9] using the results of [7] for FH singularities at a nonzero distance from each other. Finally, let us verify that (1.27) reduces to (1.13) as $|s| \to 0$ , and to (1.11) as $|s| \to \infty$ . In the former case, we substitute the small s asymptotics (9.18) of r(s) and the formula (1.13) for $D_n(f_t)$ into (1.27) and obtain by a straightforward calculation which uses the property $G(z+1) = \Gamma(z)G(z)$ of the Barnes G-function that $D_{n-1}(f_t^-)$ is given by (1.13) with n replaced by n-1 and with $\beta_2$ replaced by $\beta_2^-$ . Consider now $|s| \to \infty$ . We have $\beta' = \beta^-$ , and $\beta''_1 = \beta^-_1 - 1 = \beta_1 - 1$ , $\beta''_2 = \beta^-_2 + 1 = \beta_2$ . Using the expansion (1.8) for $D_n(f_t)$ and the second part of (1.27), we obtain (1.11) for $D_{n-1}(f_t^-)$ . In particular, this extends the validity of (1.11) (with the appropriately changed estimate for the error term): cf. Theorem 1.11 above.

Proposition 2.1. Let t > 0 and $n \in \mathbb{N}$ . Suppose that the RH problem for Y(z; n, t) is solvable. Then $D_n(f_t) \neq 0$ , and the following differential identity holds for $\alpha_k \neq 0$ , k = 1, 2: $$\frac{1}{i}\frac{d}{dt}\ln D_n(f_t) = \sum_{k=1}^{2} (-1)^k \left[ n(\alpha_k + \beta_k) - 2\alpha_k z_k \left( \frac{dY^{-1}}{dz} \widetilde{Y} \right)_{22} (z_k) \right], z_1 = e^{it}, \quad z_2 = e^{i(2\pi - t)},$$ (2.9) where $\left(\frac{dY^{-1}}{dz}\widetilde{Y}\right)_{22}(z_k) = \lim_{z \to z_k} \left(\frac{dY^{-1}}{dz}\widetilde{Y}\right)_{22}(z)$ with $z \to z_k$ non-tangentially to the unit circle. Proof. Solvability of the RH problem at t (i.e., the fact that $D_n \neq 0$ , $D_{n\pm 1} \neq 0$ ) implies the solvability in a neighborhood of t, and hence the existence (and by (2.1), (2.3), differentiability in t) of the corresponding orthogonal polynomials. We start with the identity (3.5) of [8], which, as is easy to see from the arguments in [8], holds for any parameter t of the polynomials with respect to which they are differentiable:¹ $$\frac{\partial}{\partial t} \ln D_n(f(z)) = 2n \frac{\frac{\partial \chi_n}{\partial t}}{\chi_n} + \frac{1}{2\pi} \int_0^{2\pi} \frac{\partial}{\partial t} \left( \phi_n(z) \frac{d\widehat{\phi}_n(z^{-1})}{dz} - \widehat{\phi}_n(z^{-1}) \frac{d\phi_n(z)}{dz} \right) z f(z) d\theta,$$ ^&lt;sup>1</sup>In [8], (2.10) was derived under a stronger assumption that $D_k \neq 0$ , k = 1, 2, ..., n + 1. However, a simple continuity argument shows that (2.10) holds true if only $D_n, D_{n\pm 1} \neq 0$ . where $z = e^{i\theta}$ , $f \equiv f_t$ . We would like to move the differentiation in the integral over to f noting that $$I \equiv \frac{1}{2\pi} \int_0^{2\pi} \left( \phi_n(z) \frac{d\widehat{\phi}_n(z^{-1})}{dz} - \widehat{\phi}_n(z^{-1}) \frac{d\phi_n(z)}{dz} \right) z f(z) d\theta = -2n$$ by orthogonality, and therefore $\frac{\partial I}{\partial t} = 0$ . However, because of the case $-1/2 < \operatorname{Re} \alpha_k \le 0$ for which $\frac{\partial f}{\partial t}$ is not integrable, care needs to be taken. As before, let $$C_{\varepsilon} = \bigcup_{k=1,2} \{ z \in C : |\arg z - \arg z_k| < \varepsilon \},$$ and assume that F(z) and $\frac{\partial F(z)}{\partial t}$ are analytic functions in a neighborhood of the unit circle C. Then $$\int_{C} \frac{\partial F(z)}{\partial t} f(z) dz = \int_{C \setminus C_{\varepsilon}} \frac{\partial F(z)}{\partial t} f(z) dz + \mathcal{O}(\varepsilon^{2\alpha_{1}+1}) + \mathcal{O}(\varepsilon^{2\alpha_{2}+1}), \qquad \epsilon \to 0.$$ (2.11) Note that $(z_1 = e^{it}, z_2 = e^{i(2\pi - t)})$ $$\frac{\partial}{\partial t} \int_{C \setminus C_{\varepsilon}} F(z) f(z) dz = \int_{C \setminus C_{\varepsilon}} \frac{\partial F(z)}{\partial t} f(z) dz + \int_{C \setminus C_{\varepsilon}} F(z) \frac{\partial f(z)}{\partial t} dz + i \sum_{k=1}^{2} (-1)^{k+1} z_{k} F(z_{k}) \{ f(z_{k} e^{-i\varepsilon}) - f(z_{k} e^{i\varepsilon}) \} + \mathcal{O}(\varepsilon^{2\alpha_{1}+1}) + \mathcal{O}(\varepsilon^{2\alpha_{2}+1})$$ (2.12) as $\epsilon \to 0$ . On the other hand, $$\frac{\partial}{\partial t} \int_{C \setminus C_{\varepsilon}} F(z) f(z) dz = \frac{\partial}{\partial t} \int_{C} F(z) f(z) dz + \mathcal{O}(\varepsilon^{2\alpha_{1}+1}) + \mathcal{O}(\varepsilon^{2\alpha_{2}+1})$$ (2.13) as $\epsilon \to 0$ by estimation of the integral of Ff over $C_{\varepsilon}$ . Combining (2.11), (2.12), and (2.13), we can write $$\int_{C} \frac{\partial F(z)}{\partial t} f(z) dz = \frac{\partial}{\partial t} \int_{C} F(z) f(z) dz - G, \tag{2.14}$$ $$G = \lim_{\varepsilon \to 0} \left[ \int_{C \setminus C_{\varepsilon}} F(z) \frac{\partial f(z)}{\partial t} dz + i \sum_{k=1}^{2} (-1)^{k+1} z_k F(z_k) \{ f(z_k e^{-i\varepsilon}) - f(z_k e^{i\varepsilon}) \} \right].$$ (2.15) Let us now compute $\frac{\partial f(z)}{\partial t}$ . Since $|z-z_k|^{2\alpha_k}=|2\sin\frac{\theta+(-1)^kt}{2}|^{2\alpha_k}$ , we have $$\frac{\partial}{\partial t} \ln|z - z_k|^{2\alpha_k} = (-1)^k \alpha_k \cot \frac{\theta + (-1)^k t}{2} = i(-1)^k \alpha_k \frac{z + z_k}{z - z_k}.$$ Therefore, $$\frac{\partial f(z)}{\partial t} = \sum_{k=1}^{2} (-1)^k \left( \alpha_k \frac{z + z_k}{z - z_k} + \beta_k \right) i f(z) = \sum_{k=1}^{2} (-1)^k \left( \alpha_k + \beta_k + \frac{2\alpha_k z_k}{z - z_k} \right) i f. \quad (2.16)$$ So we can write $$G = i \sum_{k=1}^{2} (-1)^k \left( (\alpha_k + \beta_k) \int_C F(z) f(z) dz + z_k \lim_{\varepsilon \to 0} \left[ 2\alpha_k \int_{C \setminus C_{\varepsilon}} \frac{F(z) f(z)}{z - z_k} dz - F(z_k) \{ f(z_k e^{-i\varepsilon}) - f(z_k e^{i\varepsilon}) \} \right] \right).$$ $$(2.17)$$ The limit in the last line is exactly $2\alpha_k$ times the regularized integral (2.6) evaluated at $z_k$ , by (2.8). By (2.14), (2.17), and (2.8) with $F(z) = (\phi_n(z) \frac{d\widehat{\phi}_n(z^{-1})}{dz} - \widehat{\phi}_n(z^{-1}) \frac{d\phi_n(z)}{dz})$ , we obtain from (2.10), $$\frac{\partial}{\partial t} \ln D_n(f(z)) = 2n \frac{\frac{\partial \chi_n}{\partial t}}{\chi_n} + i \sum_{k=1}^2 (-1)^k (2n(\alpha_k + \beta_k) - 2\alpha_k (I_{1,k} - I_{2,k})), \qquad (2.18)$$ $$I_{1,k} = \frac{1}{2\pi i} \int_C^{(r)} \frac{\phi_n(z) \frac{d}{dz} \widehat{\phi}_n(z^{-1})}{z - z_k} z_k f(z) dz,$$ $$I_{2,k} = \frac{1}{2\pi i} \int_C^{(r)} \frac{\widehat{\phi}_n(z^{-1}) \frac{d}{dz} \phi_n(z)}{z - z_k} z_k f(z) dz.$$ Let us simplify $I_{1,k}$ . Adding and subtracting $\frac{d}{dz}\widehat{\phi}_n(z^{-1})|_{z=z_k}$ from the numerator of the integrand, and observing that $$\frac{\frac{d}{dz}\widehat{\phi}_n(z^{-1}) - \frac{d}{dz}\widehat{\phi}_n(z^{-1})|_{z=z_k}}{z^{-1} - z_k^{-1}}$$ is a polynomial in $z^{-1}$ of degree n with the leading coefficient $-n\chi_n$ , we obtain by orthogonality that $$I_{1,k} = n + \frac{d}{dz}\widehat{\phi}_n(z^{-1})|_{z=z_k} \frac{1}{2\pi} \int_C^{(r)} \frac{\phi_n(z)}{z - z_k} (z - z_k + z_k) z_k f(z) d\theta$$ $$= n + z_k^2 \frac{d}{dz}\widehat{\phi}_n(z^{-1})|_{z=z_k} \frac{1}{2\pi} \int_C^{(r)} \frac{\phi_n(z)}{z - z_k} f(z) (z^{n-1} - z_k^{n-1} + z_k^{n-1}) z^{-(n-1)} d\theta$$ $$= n + z_k^{n+1} \chi_n \frac{d}{dz} \widehat{\phi}_n(z^{-1})|_{z=z_k} \widetilde{Y}_{12}(z_k). \quad (2.19)$$ Before a similar simplification of $I_{2,k}$ , it is convenient first to use the following recurrence relation (see, e.g., (2.4) in [7]): $$\chi_n \widehat{\phi}_n(z^{-1}) = \chi_{n-1} z^{-1} \widehat{\phi}_{n-1}(z^{-1}) + \widehat{\phi}_n(0) z^{-n} \phi_n(z).$$ (2.20) Substituting this into $I_{2,k}$ , and then arguing in a similar way as for $I_{1,k}$ , we obtain: $$I_{2,k} = z_k \frac{d}{dz} \phi_n(z)|_{z=z_k} \widehat{\phi}_n(0) \widetilde{Y}_{12}(z_k) - z_k \frac{dY_{11}}{dz}(z_k) \widetilde{Y}_{22}(z_k).$$ (2.21) Applying (2.20) once again to the corresponding term in (2.19) and subtracting (2.21), we obtain: $$I_{1,k} - I_{2,k} = n + z_k \frac{dY_{11}}{dz}(z_k)\widetilde{Y}_{22}(z_k)$$ $$+ \left( nY_{21}(z_k) - n\chi_n \widehat{\phi}_n(0)Y_{11}(z_k) - z_k \frac{dY_{21}}{dz}(z_k) \right) \widetilde{Y}_{12}(z_k). \quad (2.22)$$ Furthermore, $$2\frac{\frac{\partial \chi_n}{\partial t}}{\chi_n} = \frac{1}{2\pi i} \int_C f(z) \frac{\partial}{\partial t} (\phi_n(z) \widehat{\phi}_n(z^{-1})) \frac{dz}{z} = -\frac{1}{2\pi i} \int_C^{(r)} \phi_n(z) \widehat{\phi}_n(z^{-1}) \frac{\partial f(z)}{\partial t} \frac{dz}{z}$$ $$= -i \sum_{k=1}^2 (-1)^k [\alpha_k + \beta_k + 2\alpha_k I_{3,k}], \quad (2.23)$$ where $$I_{3,k} \equiv \frac{1}{2\pi i} \int_{C}^{(r)} \frac{\phi_n(z)\widehat{\phi}_n(z^{-1})}{z - z_k} z_k f(z) \frac{dz}{z}$$ (2.24) can be analyzed as the other such integrals above, in this case first adding and subtracting $\hat{\phi}_n(z_k^{-1})$ from the numerator of the integrand. We then obtain $$I_{3,k} = -1 + \chi_n \widehat{\phi}_n(z_k^{-1}) z_k^n Y_{12}(z_k).$$ Using again (2.20) gives $$I_{3,k} = -1 - Y_{21}(z_k)\widetilde{Y}_{12}(z_k) + \chi_n \widehat{\phi}_n(0)Y_{11}(z_k)\widetilde{Y}_{12}(z_k). \tag{2.25}$$ Substituting (2.25) into (2.23), and the latter with (2.22) into (2.18), we finally obtain the differential identity (2.9) as $\det Y(z) = 1$ . Remark 2.2 A differential identity for $\frac{d}{dt} \ln D_n(f(z))$ in the case when one of (or both) $\alpha_k$ 's is zero is now also easy to obtain. Either one can derive it directly using (2.17), or one can observe that both the left and the right-hand side of (2.9) are continuous in $\alpha_k$ , so that the differential identity for $\alpha_k = 0$ is obtained from (2.9) by letting $\alpha_k \to 0$ .

Proposition 3.2. There exist complex constants $c_1, c_2$ , which may depend on $\alpha_1, \alpha_2, \beta_1, \beta_2$ , but not on s, such that $$\alpha_1 \left( F_1(i;s)^{-1} F_{1,\zeta}(i;s) \right)_{22} = \frac{i}{4} \sigma(s) - \frac{i}{8} (\beta_1 + \beta_2) s + c_1, \tag{3.60}$$ $$\alpha_2 \left( F_2(-i;s)^{-1} F_{2,\zeta}(-i;s) \right)_{22} = -\frac{i}{4} \sigma(s) - \frac{i}{8} (\beta_1 + \beta_2) s + c_2. \tag{3.61}$$ Proof. By (3.20) and (3.8)-(3.12), we have $F_{j,s} = BF_j$ , j = 1, 2. Let us expand $F_j(\zeta; s)$ as $\zeta \to \pm i$ as $$F_j(\zeta;s) = F_j^{(0)}(s) \left( I + F_j^{(1)}(s)(\zeta \mp i) + \mathcal{O}(\zeta \mp i)^2 \right). \tag{3.62}$$ Substituting this into $F_{j,s} = BF_j$ , we obtain by (3.22), $$F_{j,s}^{(0)} = (B_0 \pm iB_1)F_j^{(0)}, \tag{3.63}$$ $$F_{j,s}^{(1)} = \left(F_j^{(0)}\right)^{-1} B_1 F_j^{(0)} = -\frac{i}{4} \left(F_j(\pm i; s)^{-1} \sigma_3 F_j(\pm i; s)\right). \tag{3.64}$$ Here j=1 corresponds to the upper symbol in $\pm$ or $\mp$ , and j=2 to the lower one. Also by (3.62), we have $F_i^{-1}(\pm i)F_{j,\zeta}(\pm i)=F_j^{(1)}$ , which shows that $$\alpha_j \left( F_j^{-1}(\pm i) F_{j,\zeta}(\pm i) \right)_{22,s} = -\frac{i}{4} (\mp \sigma_s + \frac{\beta_1 + \beta_2}{2})$$ (3.65) by Proposition 3.1. Integrating, we obtain (3.60)-(3.61). Remark 3.3 To find the explicit expressions for the constants $c_1, c_2$ , one can use either the large s or the small s asymptotic solution of the $\Psi$ -RH problem presented in the following sections. In this way, we obtain $$c_1 = -c_2 = \frac{i}{8}(\beta_1 + \beta_2)^2. (3.66)$$

Proposition 6.1. Let $2(\alpha_1 + \alpha_2) \notin \mathbb{N} \cup \{0\}$ . Then the function (6.12) solves the RH problem for $P_0$ . Proof. Condition (a) of the RH problem for P is satisfied by construction, as well as the jump relations on $\widehat{\Gamma}_1, \widehat{\Gamma}_4, \widehat{\Gamma}_5$ . The jump condition on $\widehat{\Gamma}_2$ follows from the definition of $P_0$ in regions II and III, and from the fact that $P_0^{(3)}$ has a jump on $\widehat{\Gamma}_2$ because of the branch cut of $\lambda^{\alpha_1 \sigma_3}$ : for $\lambda \in \widehat{\Gamma}_2$ , $$P_{0,-}(\lambda)^{-1}P_{0,+}(\lambda) = \widehat{J}_2 \left( \widetilde{G}_3^{-1} e^{-2\pi i \alpha_1 \sigma_3} \widetilde{G}_3 \right)^{-1} \widetilde{G}_3^{-1} e^{-2\pi i \alpha_1 \sigma_3} \widetilde{G}_3 = \widehat{J}_2.$$ On $\widehat{\Gamma}_3$ , we have after a similar calculation using (6.10), taking into account the branch cut of $(\lambda - s)^{\alpha_2 \sigma_3}$ , and using the value of $\widetilde{g}(\alpha_1 + \alpha_2, \beta_1 + \beta_2)$ in (4.11), $$P_{0,-}(\lambda)^{-1}P_{0,+}(\lambda) = \widehat{J}_4^{-1}\widehat{J}_5\widehat{J}_1\widehat{J}_2\widetilde{G}_3^{-1}e^{2\pi i\alpha_1\sigma_3}\widetilde{G}_3P_{0,-}^{(3)}(\lambda)^{-1}P_{0,+}^{(3)}(\lambda)$$ $$= \widehat{J}_4^{-1}\widehat{J}_5\widehat{J}_1\widehat{J}_2\widetilde{G}_2^{-1}e^{2\pi i(\alpha_1+\alpha_2)\sigma_3}\widetilde{G}_3 = \widehat{J}_3.$$ On the interval $\widehat{\Gamma}_6 = [s, 0]$ , by (6.10), (6.8), and (6.11), we obtain $$\begin{split} P_{0,-}(\lambda)^{-1}P_{0,+}(\lambda) &= \widehat{J}_{5}\widehat{J}_{1}\widehat{J}_{2}\widetilde{G}_{3}^{-1}e^{2\pi i\alpha_{1}\sigma_{3}}\widetilde{G}_{3}P_{0,-}^{(3)}(\lambda)^{-1}P_{0,+}^{(3)}(\lambda) \\ &= \begin{pmatrix} 0 & e^{\pi i(\alpha_{1}+\alpha_{2}+\beta_{1}+\beta_{2})} \\ -e^{-\pi i(\alpha_{1}+\alpha_{2}+\beta_{1}+\beta_{2})} & e^{-2\pi i(\beta_{1}+\beta_{2})} \end{pmatrix} \widetilde{G}_{3}^{-1}e^{2\pi i\alpha_{1}\sigma_{3}}\widetilde{G}_{3} \\ &\times \begin{pmatrix} 1 & 2c_{0}e^{-\pi i(3\alpha_{1}+\alpha_{2})} \\ 0 & 1 \end{pmatrix} \\ &= \widehat{J}_{6}. \end{split}$$ In fact, it is the requirement that here $P_{0,-}(\lambda)^{-1}P_{0,+}(\lambda) = \widehat{J}_6$ which fixes the value (6.11) of $c_0$ . Now we prove the matching condition $P_0(\lambda)M(\lambda)^{-1} = I + \mathcal{O}(|s|)$ on the boundary $(\partial \mathcal{U}_0) \cap \text{region III as } s \to -i0_+$ . Here we use (4.8) with $\alpha = \alpha_1 + \alpha_2$ , $\beta = \beta_1 + \beta_2$ . The branch of $\lambda^{\alpha_1 + \alpha_2} \equiv \lambda_M^{\alpha_1 + \alpha_2}$ in (4.8) was chosen with arguments between 0 and $2\pi$ . On the other hand, the branch of $\lambda^{\alpha_1} \equiv \lambda_P^{\alpha_1}$ in (6.10) was chosen with arguments between $-5\pi/4$ and $3\pi/4$ . Therefore $\lambda_M^{\alpha_1} = \lambda_P^{\alpha_1} e^{2\pi i \alpha_1}$ for $\lambda$ in region III. Taking this into account, we obtain $$P(\lambda)M(\lambda)^{-1} = L(\lambda) \begin{pmatrix} 1 & c_0 J(\lambda; s) \\ 0 & 1 \end{pmatrix} \lambda^{-\alpha_2 \sigma_3} (\lambda - s)^{\alpha_2 \sigma_3} L(\lambda)^{-1}$$ for $\lambda \in (\partial \mathcal{U}_0) \cap$ region III. Here we can (and do) choose the branch of $\lambda^{-\alpha_2}$ with arguments between $-\pi/2$ and $-5\pi/2$ , and that of $(\lambda - s)^{\alpha_2}$ with arguments between $-\pi/2$ and $3\pi/2$ . Note that this expression can be extended to the whole plane as an analytic function outside the cut [0, s]. As $s \to -i0_+$ , we obtain uniformly on the boundary $\partial \mathcal{U}_0$ : $$P(\lambda)M(\lambda)^{-1} = I + \Delta_1(\lambda) + \mathcal{O}(|s|^2) + \mathcal{O}(|s|^{2+4(\alpha_1 + \alpha_2)}) = I + o(1), \quad \lambda \in \partial \mathcal{U}_0, \quad (6.13)$$ where $$\Delta_{1}(\lambda) = -\alpha_{2} \frac{s}{\lambda} L(\lambda) \sigma_{3} L(\lambda)^{-1} - \frac{c_{0}}{\pi \lambda} |s|^{1+2(\alpha_{1}+\alpha_{2})} \frac{\Gamma(1+2\alpha_{1})\Gamma(1+2\alpha_{2})}{\Gamma(2+2(\alpha_{1}+\alpha_{2}))} L(\lambda) \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix} L(\lambda)^{-1}. \quad (6.14)$$ Here we used (6.8) and the connection between Beta and Gamma functions. Alternatively, we can use (6.7). Finally, we need to prove condition (d) of the RH problem. Since $\widehat{\Psi}$ and $P_0$ have the same jump relations, the possible singularities of $\widehat{\Psi}(\lambda)P_0(\lambda)^{-1}$ at 0 and s are isolated, and it is easily verified by the construction of $P_0$ and the behavior of $\widehat{\Psi}$ at 0 and s that the singularities cannot be essential. Therefore, to conclude that the singularities are removable, it suffices to check that $\widehat{\Psi}(\lambda)P_0(\lambda)^{-1}$ is $o(|\lambda|^{-1})$ as $\lambda \to 0$ in region III, and that it is $o(|\lambda - s|^{-1})$ as $\lambda \to s$ in region III. Let us consider the behavior of $\widehat{\Psi}(\lambda)P_0(\lambda)^{-1}$ as $\lambda \to 0$ in region III. From the conditions (3.8), (3.15), we obtain as $\lambda \to 0$ in region III: $$\widehat{\Psi}(\lambda) = \widehat{F}_1(\lambda)\lambda^{\alpha_1\sigma_3} \begin{pmatrix} 1 & \ell(\lambda)e^{-i\pi(\alpha_1-\beta_1-\alpha_2+\beta_2)} \\ 0 & 1 \end{pmatrix},$$ $$\widehat{F}_1(\lambda) = e^{-\frac{s}{4}\sigma_3}e^{-i\frac{\pi}{2}(\alpha_1-\beta_1-\alpha_2+\beta_2)\sigma_3}F_1\left(\frac{2}{|s|}\lambda+i\right)e^{i\frac{\pi}{2}(\alpha_1-\beta_1-\alpha_2+\beta_2)\sigma_3}\left(\frac{2}{|s|}\right)^{\alpha_1\sigma_3},$$ (6.15) where $\ell = g$ , with g given by (3.9) for $2\alpha_1 \neq 0, 1, \ldots$ ; and $\ell = g_{int} \ln(2\lambda/|s|)$ , with $g_{int}$ given by (3.16) for $2\alpha_1 = 0, 1, \ldots$ Multiplying (6.15) on the right by $P_0(\lambda)^{-1}$ and substituting (6.10), we obtain after a straightforward analysis of (6.8) that $$\widehat{\Psi}(\lambda)P_0(\lambda)^{-1} = \widehat{F}_1(\lambda) \begin{pmatrix} 1 & \mathcal{O}(|\lambda|^{2\alpha_1}) + \mathcal{O}(|\ln \lambda|) + \mathcal{O}(1) \\ 0 & 1 \end{pmatrix} L(\lambda)^{-1}, \tag{6.16}$$ as $\lambda \to 0$ in region III. Since $\widehat{F}_1$ and L are analytic at 0 and Re $\alpha_1 > -1/2$ , this implies that $\widehat{\Psi}(\lambda)P_0(\lambda)^{-1}$ is analytic at 0. In a similar way we obtain $$\widehat{\Psi}(\lambda)P_0(\lambda)^{-1} = \widehat{F}_2(\lambda) \begin{pmatrix} 1 & \mathcal{O}(|\lambda|^{2\alpha_2}) + \mathcal{O}(|\ln \lambda|) + \mathcal{O}(1) \\ 0 & 1 \end{pmatrix} L(\lambda)^{-1}, \tag{6.17}$$ as $\lambda \to s$ in region III. Hence $\widehat{\Psi}(\lambda)P_0(\lambda)^{-1}$ is analytic at s. Remark 6.2 The representation (6.7) allows us to obtain the full expansion of $P_0(\lambda)$ near its singularities without much effort (cf. [5]). The exact cancellation of singular parts of $P_0(\lambda)$ and $\widehat{\Psi}(\lambda)$ at 0, s, in the expression $\widehat{\Psi}(\lambda)P_0(\lambda)^{-1}$ gives an alternative way to fix the value (6.11) of the constant $c_0$ . Consider, for example, the case of $\lambda$ near 0 and $2\alpha_1 \neq 0, 1, 2...$ Then we can use the following standard transformation of the hypergeometric function: $$F(1, 1 + 2\alpha_1, 2 + 2(\alpha_1 + \alpha_2), z) = -\frac{\pi}{\sin 2\pi\alpha_1} \frac{\Gamma(2 + 2(\alpha_1 + \alpha_2))}{\Gamma(1 + 2\alpha_1)\Gamma(1 + 2\alpha_2)}$$ $$\times (e^{i\pi}z^{-1})^{1+2\alpha_1} \left(1 - \frac{1}{z}\right)^{2\alpha_2}$$ $$+ \frac{1 + 2(\alpha_1 + \alpha_2)}{2\alpha_1} e^{i\pi}z^{-1} F(1, -2(\alpha_1 + \alpha_2), 1 - 2\alpha_1, 1/z),$$ (6.18) to write $J(\lambda; s)$ in the form $$J(\lambda;s) = J_{sing}(\lambda) + J_{an}(\lambda), \qquad J_{sing}(\lambda) = \frac{e^{i\pi(3\alpha_1 - \alpha_2)}}{i\sin 2\pi\alpha_1} \lambda^{2\alpha_1} (\lambda - s)^{2\alpha_2}, \qquad (6.19)$$ $$J_{an}(\lambda) = -\frac{1}{i\pi} |s|^{2(\alpha_1 + \alpha_2)} \frac{\Gamma(2\alpha_1)\Gamma(1 + 2\alpha_2)}{\Gamma(1 + 2(\alpha_1 + \alpha_2))} F\left(1, -2(\alpha_1 + \alpha_2), 1 - 2\alpha_1, \frac{\lambda}{s}\right), \qquad (6.20)$$ with the branch of $\lambda^{\alpha_1}$ corresponding to the arguments between $-\pi/2$ and $-5\pi/2$ , and the branch of $(\lambda - s)^{\alpha_2}$ , to the arguments between $3\pi/2$ and $-\pi/2$ . Since $J_{an}(\lambda)$ is analytic at $\lambda = 0$ , this representation shows the form of the singularity of $J(\lambda)$ at zero. Note that in region III, the branches in (6.19) coincide with those in (6.10). We now write $P_0$ as $\lambda \to 0$ in region III in the form $$P_{0}(\lambda) = L(\lambda) \begin{pmatrix} 1 & c_{0}J_{an}(\lambda) \\ 0 & 1 \end{pmatrix} (\lambda - s)^{\alpha_{2}\sigma_{3}} \begin{pmatrix} 1 & c_{0}J_{sing}(\lambda)(\lambda - s)^{-2\alpha_{2}} \\ 0 & 1 \end{pmatrix} \times \lambda^{\alpha_{1}\sigma_{3}} e^{2\pi i \alpha_{1}\sigma_{3}} \begin{pmatrix} 1 & \widetilde{g}(\alpha_{1} + \alpha_{2}, \beta_{1} + \beta_{2}) \\ 0 & 1 \end{pmatrix}, \quad (6.21)$$ where $\widetilde{g}(\alpha, \beta)$ is given by (4.11). Comparing this expression with (6.15), we see that the condition of analyticity of $\widehat{\Psi}(\lambda)P_0(\lambda)^{-1}$ at zero is the condition of vanishing of the term with $\lambda^{2\alpha_1}$ in $\widehat{\Psi}(\lambda)P_0(\lambda)^{-1}$ , which is $$ge^{-i\pi(\alpha_{1}-\beta_{1}-\alpha_{2}+\beta_{2})} - \widetilde{g}(\alpha_{1}+\alpha_{2},\beta_{1}+\beta_{2})$$ $$= c_{0}J_{sing}(\lambda)(\lambda-s)^{-2\alpha_{2}}\lambda^{-2\alpha_{1}}e^{-4\pi i\alpha_{1}} = c_{0}\frac{e^{-i\pi(\alpha_{1}+\alpha_{2})}}{i\sin 2\pi\alpha_{1}}.$$ Solving this condition for $c_0$ , we again obtain (6.11). We now construct the parametrix in the remaining case $2(\alpha_1 + \alpha_2) \in \mathbb{N} \cup \{0\}$ . (Note, in particular, that the constant $c_0$ in (6.11) is not defined in this case.) Set $$\widetilde{J}(\lambda;s) = \frac{1}{2} \frac{\partial}{\partial \alpha_1} J(\lambda;s) = \frac{1}{\pi i} \int_s^0 \frac{|\xi|^{2\alpha_1} |\xi - s|^{2\alpha_2} \ln |\xi|}{\xi - \lambda} d\xi, \tag{6.22}$$ and then set $$\widetilde{P}_{0}^{(3)}(\lambda) = \widetilde{L}(\lambda) \begin{pmatrix} 1 & e_{1}\widetilde{J}(\lambda;s) + e_{2}J(\lambda;s) \\ 0 & 1 \end{pmatrix} \lambda^{\alpha_{1}\sigma_{3}}(\lambda - s)^{\alpha_{2}\sigma_{3}} e^{2\pi i \alpha_{1}\sigma_{3}} \begin{pmatrix} 1 & m(\lambda) \\ 0 & 1 \end{pmatrix}, (6.23)$$ where $$e_1 = \frac{i}{\pi} e^{-i\pi(\alpha_1 + \alpha_2)} \sin \pi (\alpha_1 + \alpha_2 + \beta_1 + \beta_2) \sin 2\pi \alpha_1, \tag{6.24}$$ $$e_2 = \frac{1}{2} \left( i\pi c_1 + e^{i\pi(-2\alpha_1 + \beta_1 + \beta_2)} - (-1)^{2(\alpha_1 + \alpha_2)} e^{i\pi(\beta_1 - \beta_2)} \right), \tag{6.25}$$ the matrix $\widetilde{L}$ and $m(\lambda)$ are as in (4.12) and (4.13), respectively, with $\alpha = \alpha_1 + \alpha_2$ , $\beta = \beta_1 + \beta_2$ . $\beta = \beta_1 + \beta_2.$ With $\widetilde{P}_0^{(3)}(\lambda)$ given by (6.23) set $$P_{0}(\lambda) = \widetilde{P}_{0}^{(3)}(\lambda), \quad \text{in region III,}$$ $$P_{0}(\lambda) = \widetilde{P}_{0}^{(3)}(\lambda) \begin{pmatrix} 1 & -m(\lambda) \\ 0 & 1 \end{pmatrix} e^{-2\pi i \alpha_{1} \sigma_{3}} \begin{pmatrix} 1 & m(\lambda) \\ 0 & 1 \end{pmatrix} \widehat{J}_{2}^{-1}$$ $$\times \begin{cases} \widehat{J}_{2}^{-1}, & \text{in region II,} \\ \widehat{J}_{2}^{-1} \widehat{J}_{1}^{-1}, & \text{in region I,} \\ (-1)^{2(\alpha_{1} + \alpha_{2})} \widehat{J}_{3}^{-1}, & \text{in region IV,} \\ (-1)^{2(\alpha_{1} + \alpha_{2})} \widehat{J}_{3}^{-1} \widehat{J}_{4}^{-1}, & \text{in region V.} \end{cases}$$ (6.26) Then the jump conditions hold (the jump condition on $\widehat{\Gamma}_6$ fixes the values (6.24), (6.25) of the constants $c_1$ and $c_2$ ). One verifies conditions (c) and (d) in a similar way as above. In particular, we obtain $$P_0(\lambda)M(\lambda)^{-1} = I + \mathcal{O}(|s\ln|s||), \qquad \lambda \in \partial \mathcal{U}_0, \qquad s \to -i0_+.$$ (6.27) Thus, we have

Proposition 6.3. Let $2(\alpha_1 + \alpha_2) \in \mathbb{N} \cup \{0\}$ . Then the function (6.26) solves the RH problem for $P_0$ .

Proposition 7.1. Let $\tilde{n} = \min\{n, \sqrt{n/t}\}$ , and assume that $|||\beta||| < 1$ . We have $$P(z)N(z)^{-1} = \widetilde{n}^{-(\beta_1 + \beta_2)\sigma_3} (I + \mathcal{O}(t(nt)^{-1 + |||\beta|||})) \widetilde{n}^{(\beta_1 + \beta_2)\sigma_3}, \quad n \to \infty, \ z \in \partial \mathcal{U}, \ (7.24)$$ uniformly for $0 < t < t_0$ with $t_0$ sufficiently small and uniformly in $z \in \partial \mathcal{U}$ . Proof. Let us first consider the case where $c_0 \leq nt \leq C_0$ , with some $c_0 > 0$ small and some $C_0 > 0$ large. The constants $c_0$ , $C_0$ will be fixed below. Then $|\frac{1}{t} \ln z| > \delta n$ for $z \in \partial \mathcal{U}$ , so s = -2int remains bounded and bounded away from zero, and by (7.21) and (7.19), we have $$P(z)N(z)^{-1} = E(z)(I + \mathcal{O}(n^{-1}))\widehat{P}^{(\infty)}(\frac{1}{t}\ln z)z^{-\frac{n}{2}\sigma_3}W(z)N(z)^{-1}, \quad z \in \partial \mathcal{U}, \quad n \to \infty.$$ (7.25) Recall that we assume that the problem for $\Psi$ is solvable for $c_0 \leq nt \leq C_0$ . Therefore, by general properties of Painlevé RH problems, the estimate for the error term here is valid uniformly for all $c_0 \leq nt \leq C_0$ . By (7.9) and (7.22), we obtain $$z^{-\frac{n}{2}\sigma_3}W(z)N(z)^{-1} = (\mathcal{D}_{in,t}(z)\mathcal{D}_{out,t}(z))^{\frac{1}{2}\sigma_3}\sigma_1, \quad \text{for } z \in \partial \mathcal{U}.$$ (7.26) Substituting this into (7.25), we see the reason for the definition of E in (7.23). Furthermore, we set $$\widehat{E}(z) = n^{(\beta_1 + \beta_2)\sigma_3} E(z) \tag{7.27}$$ so that $\widehat{E}$ is bounded in n (uniformly for $z \in \partial \mathcal{U}$ and uniformly for $t < t_0$ ): this follows easily from (7.20) and (3.7). In particular, $$\widehat{E}(z) = \sigma_1 \left( \mathcal{D}_{in,t}(z) \mathcal{D}_{out,t}(z) \right)^{-\frac{1}{2}\sigma_3} (\ln z - it)^{\beta_1 \sigma_3} (\ln z + it)^{\beta_2 \sigma_3}, \qquad -1 < \zeta < 1.$$ (7.28) We can now write (7.25) as follows: $$P(z)N(z)^{-1} = n^{-(\beta_1+\beta_2)\sigma_3} \widehat{E}(z)(I + \mathcal{O}(n^{-1}))\widehat{E}(z)^{-1} n^{(\beta_1+\beta_2)\sigma_3}$$ = $\widetilde{n}^{-(\beta_1+\beta_2)\sigma_3} (I + \mathcal{O}(n^{-1}))\widetilde{n}^{(\beta_1+\beta_2)\sigma_3}, \quad z \in \partial \mathcal{U}, \quad n \to \infty.$ (7.29) This proves that (7.24) holds (uniformly) for $c_0 \leq nt \leq C_0$ , $z \in \partial \mathcal{U}$ . Next, suppose $C_0 < nt \le \omega(n)$ . In this case we cannot use the expansion (7.19) since the argument s of $\Psi_1(s)$ is not bounded. Instead we use the large |s| = 2nt asymptotics for $\Psi$ . For that, we need $C_0$ to be sufficiently large. The asymptotics will be valid for the whole region $C_0 < nt < t_0$ . Note that, for $z \in \partial \mathcal{U}$ and t sufficiently small (i.e., $t < t_0$ ), $\zeta = \frac{1}{t} \ln z$ is sufficiently large in absolute value to lie outside of the regions $\mathcal{U}_1$ and $\mathcal{U}_2$ defined in Section 5.3. By (7.21), (5.1), (5.17), and (5.18), we have $$P(z)N(z)^{-1} = E(z)\Phi(\frac{1}{t}\ln z; -2int)W(z)N(z)^{-1}$$ $$= E(z)(nt)^{-\frac{1}{2}(\beta_1+\beta_2)\sigma_3}R(\frac{1}{t}\ln z; -2int)(nt)^{\frac{1}{2}(\beta_1+\beta_2)\sigma_3}$$ $$\times \widehat{P}^{(\infty)}(\frac{1}{t}\ln z)z^{-\frac{n}{2}\sigma_3}W(z)N(z)^{-1}, \quad z \in \partial \mathcal{U}.$$ (7.30) Using (7.27), and (7.26), we obtain for $z \in \partial \mathcal{U}$ : $$P(z)N(z)^{-1} = n^{-(\beta_1 + \beta_2)\sigma_3} \widehat{E}(z)(nt)^{-\frac{1}{2}(\beta_1 + \beta_2)\sigma_3} R(\frac{1}{t} \ln z; -2int)(nt)^{\frac{1}{2}(\beta_1 + \beta_2)\sigma_3} \times \widehat{E}(z)^{-1} n^{(\beta_1 + \beta_2)\sigma_3}$$ $$= \left(\frac{n}{t}\right)^{-\frac{1}{2}(\beta_1 + \beta_2)\sigma_3} \widehat{E}(z)R(\frac{1}{t} \ln z; -2int)\widehat{E}(z)^{-1} \left(\frac{n}{t}\right)^{\frac{1}{2}(\beta_1 + \beta_2)\sigma_3}.$$ $$(7.32)$$ Therefore, by (5.25), we have (uniformly for $C_0/n < t < t_0, z \in \partial \mathcal{U}$ ) $$P(z)N(z)^{-1} = \widetilde{n}^{-(\beta_1 + \beta_2)\sigma_3} (I + \mathcal{O}(t(nt)^{-1 + ||\beta|||})) \widetilde{n}^{(\beta_1 + \beta_2)\sigma_3}. \quad n \to \infty, \quad z \in \partial \mathcal{U}.$$ (7.33) If $nt < c_0$ , we can use the small |s| asymptotics for $\Psi(\zeta; s)$ for large values of $\zeta = \frac{1}{t} \ln z$ . We need to consider this case separately from $c_0 \le nt \le C_0$ since s = 0 is a branching point for the Painlevé functions. By (6.1), (6.28), (6.32), and (4.6), we have for $z \in \partial \mathcal{U}$ and $\arg z \notin (t, 2\pi - t)$ , $$\Psi(\frac{1}{t}\ln z; -2int) = (I + \mathcal{O}(n^{-1})) \, \widehat{P}^{(\infty)}(\frac{1}{t}\ln z) z^{-\frac{n}{2}\sigma_3}.$$ (7.34) This implies that $$P(z)N(z)^{-1} = \tilde{n}^{-(\beta_1 + \beta_2)\sigma_3} (I + \mathcal{O}(n^{-1}))\tilde{n}^{(\beta_1 + \beta_2)\sigma_3}, \quad \text{as } n \to \infty.$$ (7.35) For $2\pi - t < \arg z < 2\pi$ and $0 < \arg z < t$ , the same estimate can be proved similarly. [ For later use, we note that $$\widehat{E}^{-1}(z)\widehat{E}'(z) = h(z)\sigma_3,\tag{7.36}$$ $$h(z) = -\frac{1}{2} \sum_{j=1}^{+\infty} j V_j z^{j-1} + \frac{1}{2} \sum_{j=-1}^{-\infty} j V_j z^{j-1} - \frac{\beta_1}{z - e^{it}} + \frac{\beta_1}{z \ln z - itz} - \frac{\beta_2}{z - e^{-it}} + \frac{\beta_2}{z \ln z + itz} - \frac{\alpha_1 - \beta_1 + \alpha_2 - \beta_2}{2z}.$$ (7.37)

Proposition 8.1. Suppose that $\operatorname{Re} \alpha_1$ , $\operatorname{Re} \alpha_2$ , $\operatorname{Re} (\alpha_1 + \alpha_2) > -\frac{1}{2}$ , $\alpha_k \pm \beta_k \neq -1, -2, \ldots$ , k = 1, 2, $(\alpha_1 + \alpha_2) \pm (\beta_1 + \beta_2) \neq -1, -2, \ldots$ , and that $|||\beta||| < 1$ . Let $\sigma(s)$ be the solution to (1.14) analyzed above, and let $\mathcal{P}$ be an open subset of the s = -2int plane $\mathbb{C}$ containing the set $\Omega$ of all the (finitely many) nonzero points where the $\Psi$ -RH problem is not solvable. Let $\omega(x) \equiv \omega(x; |||\beta|||)$ be a positive, smooth function for x sufficiently large, with the following behavior: $$\omega(n) \to \infty, \quad \omega(n) = o(n^{\varepsilon}), \qquad \varepsilon = \min\left\{1, \frac{2}{1 + 2|||\beta|||}\right\}, \qquad as \ n \to \infty. \quad (8.1)$$ There holds the following asymptotic expansion: $$\frac{1}{i}\frac{d}{dt}\ln D_n(f_t) = n(\beta_2 - \beta_1) + d_1(t; \alpha_1, \beta_1, \alpha_2, \beta_2) + d_2(n, t; \alpha_1, \beta_1, \alpha_2, \beta_2) + d_3(t; \alpha_1, \beta_1, \alpha_2, \beta_2) + \mathcal{E}_{n,t}, \quad (8.2)$$ as $n \to \infty$ , where $\mathcal{E}_{n,t}$ is such that $$\left| \int_0^t \mathcal{E}_{n,x} dx \right| = \mathcal{O}(\omega(n)^{-1+|||\beta|||}) + \mathcal{O}(n^{-2}\omega(n)^{1+2|||\beta|||}) = o(1), \tag{8.3}$$ uniformly for $0 < t < t_0$ and $-2int \in \mathbb{C} \setminus \mathcal{P}$ (the path of integration in (8.3) avoids the points of $\Omega$ ), and where $$d_1(t; \alpha_1, \beta_1, \alpha_2, \beta_2) = -\alpha_1 \sum_{j \neq 0} j V_j e^{ijt} + \alpha_2 \sum_{j \neq 0} j V_j e^{-ijt} + (\alpha_2 - \alpha_1)(\beta_1 + \beta_2)$$ $$+ i(\beta_1 + \beta_2) \sum_{j=1}^{+\infty} j(V_j - V_{-j}) \sin(jt), \tag{8.4}$$ $$d_2(n, t; \alpha_1, \beta_1, \alpha_2, \beta_2) = ((\beta_1 + \beta_2)^2 - 4\alpha_1\alpha_2) \frac{\cos t}{2i\sin t} + \frac{1}{it}\sigma(-2int), \tag{8.5}$$ $$d_{3}(t; \alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}) = 2\sigma_{s} \left[ -\sum_{j=1}^{+\infty} j(V_{j} + V_{-j}) \cos(jt) + \frac{\beta_{1} - \beta_{2}}{2i} \left( \frac{\cos t}{\sin t} - \frac{1}{t} \right) - \alpha_{1} - \alpha_{2} \right].$$ (8.6) Proof. In the derivation below, we assume $\alpha_k \neq 0$ , k = 1, 2, so that we can use Proposition 2.1. Once (8.2) is proved under this assumption, the general result follows immediately from the uniformity of the error term in $\alpha_1$ , $\alpha_2$ . This uniformity is easy to verify from the constructions above. Alternatively, one can consider the case $\alpha_k = 0$ separately using the corresponding differential identity: see Remark 2.2. For simplicity of the notation, we also assume below that $\operatorname{Re} \alpha_k \geq 0$ , k=1,2 (in this case, $\widehat{Y} = Y$ in the Proposition 2.1). The extension to the case $-1/2 < \operatorname{Re} \alpha_k < 0$ is an easy exercise. The plan of the proof is as follows. First, we express the differential identity of Proposition 2.1 in terms of the parametrices of the previous section (separately for the regions $0 < t \le \omega(n)/n$ , $\omega(n)/n < t < t_0$ ) and estimate the error terms. The error term estimation is especially involved in the latter region. To show that (8.3) is o(1), we use, in particular, large oscillations of $\mathcal{E}_{n,t}$ . This difficulty is caused by the presence of $\beta$ -singularities, the situation in the case of $\beta_k = 0$ , k = 1, 2, and even in the case $|||\beta||| < 1/2$ , is simpler. Second, we compute the leading asymptotic terms in the differential identity from the parametrices.