Gap probability of the circular unitary ensemble with a Fisher-Hartwig singularity and the coupled Painlevé V system
Shuai-Xia Xu Yu-Qiu Zhao
math-ph math.MP 33E17, 34M55, 41A60



A new and consistent well model for one-phase flow in anisotropic porous media using a distributed source model
    [
    Received; accepted; 
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Abstract  
We consider the circular unitary ensemble with a Fisher-Hartwig singularity of both jump type and root type at z=1. A rescaling of the ensemble at the Fisher-Hartwig singularity leads to the  confluent hypergeometric kernel. By studying the asymptotics of the Toeplitz determinants, we show that the probability of there being  no eigenvalues in a symmetric arc  about the singularity on the unit circle for a random matrix in the ensemble can be explicitly evaluated via an integral of the Hamiltonian of the coupled Painlevé V system in dimension four.  This leads to a Painlevé-type representation of the   confluent hypergeometric-kernel  determinant. Moreover, the large gap asymptotics, including the constant terms, are  derived by evaluating the total integral of the Hamiltonian. In particular, we reproduce the large gap asymptotics of the confluent hypergeometric-kernel determinant obtained by Deift, Krasovsky and Vasilevska,  and the sine-kernel determinant as a special case, including the constant term conjectured earlier by Dyson.


  .5cm
 2010 Mathematics subject classification: 33E17; 34M55; 41A60



 Keywords:Random matrices; Tracy-Widom distribution; Painlevé equations; Riemann-Hilbert problem; constant problem; Deift-Zhou method.
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§ INTRODUCTION AND STATEMENT OF RESULTS
 


One of the most celebrated results in random matrix theory is the universality of local statistics of eigenvalues for large random matrices.  Well-known universality classes  arise in the unitary invariant random matrices in the bulk, and at the soft edges and hard edges   <cit.>.  Accordingly, for large unitary random matrices, the eigenvalues near a regular point in the bulk, near the soft edge and the hard edge, form determinantal point processes  associated with the sine kernel, Airy kernel and Bessel kernel, respectively; see  <cit.> and <cit.>.  These three determinantal point processes are now well understood. A significant feature   is   that the gap probabilities in these determinantal point processes can be expressed in terms of the Painlevé transcendents  and their associated Hamiltonians; see <cit.>.  The large gap asymptotics, including the non-trivial constant terms, are also worked out; see <cit.>.

Another basic universality class in the unitary  random matrix ensembles is the confluent hypergeometric-kernel  determinantal point process, having been used to  describe  the statistics of eigenvalues  near a Fisher-Hartwig singularity in the spectrum; see  <cit.>.   For example, we consider the following  Gaussian unitary ensemble  with a Fisher-Hartwig singularity at the origin, defined by the
 joint probability density function
 
    p_n(λ_1,...,λ_n)=1/Z_n∏_j=1^n e^- λ_j^2|λ_j|^2αχ(λ_j)∏_j<k(λ_j-λ_k)^2,

where
Z_n is the normalization constant, the parameter  α>-1/2, and the function
  
    χ(x)={[  e^iπβ,    x<0,; e^-iπβ,    x>0, ].

with iβ∈ℝ.  The simultaneous occurrence of a  jump type singularity as χ(x), and  a root type singularity |x|^2α,  indicates a Fisher-Hartwig singularity at the origin; cf. <cit.>.
 For α∈ℕ and β=0, the ensemble  is a degenerate GUE of size n + α with α   eigenvalues  located at the origin, which was considered by Chen and Feigin in  <cit.>. If α=0 and iβ∈ℝ, the ensembles can be realized as a  conditional GUE.  More precisely,  one obtains a thinned process of eigenvalues by deleting  each of the eigenvalues of the original  GUE  independently with probability p=e^-2iβ for iβ>0 (while p=e^2iβ if iβ<0); see <cit.>. One may interpret the remaining and removed eigenvalues
  as observed and unobserved particles, respectively.  Knowing the information that the particles observed are all negative when iβ>0 (or positive if iβ<0),  the eigenvalue distribution of the conditional GUE  is given  by (<ref>); see <cit.>. The asymptotics of the partition function and the recurrence coefficients of the  orthogonal polynomials  associated with the Gaussian weight with a Fisher-Hartwig singularity at the origin have been obtained by Its and Krasovsky in <cit.>. The asymptotic  analysis of the orthogonal polynomials therein  implies that the scaling limit of the correlation kernel near the origin can be expressed in terms of the confluent hypergeometric kernel. For β=0, the kernel is reduced to the Bessel kernel of the first kind,   obtained earlier by Kuijlaars and Vanlessen in <cit.>.

  The confluent hypergeometric kernel also arises in the Dyson circular unitary ensemble with a Fisher-Hartwig singularity
  
    p_n(θ_1,⋯,θ_n)=1/Z_n'∏_j=1^n |e^iθ_j-1|^2αe^iβ(θ_j -π)∏_j<k  |e^iθ_j-e^iθ_k  |^2,   θ_1,...,θ_n∈(0, 2π),
 with real parameter α>-1/2 and  pure imaginary parameter β.
Then, it is direct to see that the  weight function |e^iθ-1|^2αe^iβ(θ -π) is positive for θ∈(0,2π) and possesses  a Fisher-Hartwig singularity of both root type and jump type at θ=0. With the normalization constant Z_n',    (<ref>) is made a probability measure. We denote by C_t an arc of the  unit circle, oriented counterclockwise, that is,
 
    C_t={e^iθ: t ⩽θ⩽ 2π-t }     0< t<π,

 and define the following positive weight function w(z;t) on C_t
  
    w(z;t)=|z-1|^2αz^βe^-π iβ,     z=e^iθ∈ C_t,

with real parameter α>-1/2 and  pure imaginary parameter β.
Let D_n(t) denote the Toeplitz determinant with the  weight function (<ref>), namely

    D_n(t)   =(1/2π∫_0^2πw(e^iθ;t)e^-(j-k)iθdθ)_j,k=0^n-1
       =1/(2π)^nn!∫_0^2π⋯∫_0^2π∏_j=1^n w(e^iθ_j;t)∏_j<k | e^iθ_j-e^iθ_k |^2 ∏_j=1^ndθ_j,

where the second expression is known as  Heine's multiple integral representation <cit.>.
Then, the probability that all eigenvalues of the random matrix in the ensemble (<ref>) are contained in the arc C_t   can be expressed  as

    ({θ_j∈(t,2π-t): 1⩽ j⩽ n})=D_n(t)/D_n(0).

The circular ensemble  (<ref>) was introduced by Deift, Krasovsky and Vasilevska
 in <cit.> to derive the large gap asymptotics for the Fredholm determinant of the  confluent hypergeometric kernel. It was shown that the scaling limit of the correlation kernel near z=1 is the confluent hypergeometric kernel
 
    K^(α,β)(u,v)=1/2π iΓ(1+α+β)Γ(1+α-β)/Γ(1+2α)^2A(u)B(v)-A(v)B(u)/u-v,

where
 
    A(x)=χ(x)^1/2|2x|^αe^-ixψ(1+α+β,1+2α,2ix),    B(x) =A(x),

 the confluent hypergeometric function <cit.>
 
    ψ(a,b,z)=1+∑_k=1^∞a(a+1)⋯ (a+k-1)/b(b+1)⋯(b+k-1)z^k/k!,

and χ(x)={[  e^iπβ,    x<0,; e^-iπβ,    x>0, ]. as defined in (<ref>). If α=β=0, the kernel is reduced to the classical sine kernel,
        
    K^(0,0)(u,v)=K_sin(u,v)=
    sin (u-v)/π(u-v) .

Thus, for n large, the gap probability (<ref>) can be expressed in terms of the Fredholm determinant

    lim_n→∞D_n(2t/n)/D_n(0)=(I-K^(α,β)_t),

where K^(α,β)_t is the integral operator on L^2(-t,t), t∈(0,π),  with the confluent hypergeometric kernel (<ref>); see <cit.>.   In <cit.>, the large-n asymptotics of the logarithmic derivative of  the Toeplitz determinant  D_n(t_n) were derived, uniformly
for the parameter 2t_0/n  < t _n<π
with   t_0   sufficiently large. The
boundary condition for D_n(t) as t→π was obtain by comparing it with the Hankel determinant associated with the Legendre polynomials, of which the large-n asymptotics was derived before in <cit.>.
 Therefore, asymptotic approximation  of the Toeplitz determinant,  uniformly for  2t_0/n  < t _n<π with t_0 big enough, was obtained by
 integration. Then, recalling the relation (<ref>), the large gap asymptotics of the determinant of confluent hypergeometric kernel as t→∞ was readily worked out. The  constant term, namely the Widom-Dyson constant, was expressed in terms of the Barnes G-functions therein.


 Recently, the studies of the asymptotics of Toeplitz determinant  with Fisher-Hartwig singularities are of
 great interest. In <cit.>, the asymptotics of Toeplitz determinant  for a general weight function on the circle with several Fisher-Hartwig singularities were derived and   the Barnes G-functions were involved.  A nice review of the history of the asymptotics of Toeplitz determinants, with   applications in the Ising model,   can be found in <cit.>. In <cit.>,  Claeys, Its and Krasovsky studied the emergence of a Fisher-Hartwig singularity for the Toeplitz determinant. They obtained transition type asymptotics of the  Toeplitz determinant between the  case with no singularity and the case with one  Fisher-Hartwig singularity. Later, the transition asymptotics as two Fisher-Hartwig  singularities merge into  one Fisher-Hartwig  singularity was considered by  Claeys and Krasovsky in <cit.>.  Both of the transition asymptotics are described by the Painlevé V transcendents with different
 boundary behaviors. Applications in two-dimensional Ising models and random matrix theory were also considered  therein.

From the relation (<ref>),  the large-n asymptotics of the  Toeplitz determinant
D_n(2t/n) defined in (<ref>) for bounded t will give  the determinant of the confluent hypergeometric kernel on (-t,t). Thus one obtains in the unitary ensembles (<ref>) the  probability that the symmetric interval around the  Fisher-Hartwig singularity is free of eigenvalues.  For β=0, the  kernel was reduced to the Bessel kernel of the first kind and the determinant can be expressed  in terms of the Hamiltonian associated with the Painlevé V equation; see <cit.>.  It was observed by Borodin and Deift  <cit.> that the determinant of the confluent hypergeometric kernel, restricted to one side interval (0,t), should be related to a solution to the Painlevé V equation.
Since the confluent hypergeometric kernel is not symmetric, thus the determinant on the interval (-t,t)  can not be obtained from that on the one side interval (0,t).  Moreover, it is interesting to note that as 2t/n→ 0, in (<ref>), the two  Fisher-Hartwig singularities of jump type at θ=2t/n and θ= 2π-2t/n are merging together  to the root type singularity |e^iθ-1|^2α. The result of such a process is a new Fisher-Hartwig singularity of both jump-type and the root-type singularity in the weight function in (<ref>).  The transition is different from the ones considered in <cit.> as mentioned above.  It is of interest to study the new transition asymptotics of the  Toeplitz determinant from this point of view.

More precisely, this paper is devoted to the study of the determinant of the confluent hypergeometric kernel on (-t,t) both for finite t and the asymptotics as t→∞. Firstly, we  derive the  large-n asymptotics of the Toeplitz determinant D_n(2t/n) defined in (<ref>) for bounded t.  Using the relation (<ref>) with the Toeplitz determinant, we obtain an integral representation of the determinant in terms of the Hamiltonian of a coupled Painlevé V system. We further study the existence of solutions to the system and the asymptotics of the solutions.  Secondly,  we evaluate the total integral of the  Hamiltonian, which then gives us the  large gap asymptotics of the determinant, including the constant terms.


 The Hamiltonian  integral is  closely related to the Jimbo-Miwa-Ueno tau-function, since the logarithmic derivative of the tau-function is
 the Hamiltonian; see <cit.>.  The tau-functions of Painlevé equations play important roles in random matrix theory, two-dimensional quantum gravity,  Ising models  and many other areas of  mathematical physics; see <cit.>.  Usually, it is a hard problem to determine the
 constant term in the asymptotics of the tau-functions.    Recently,  Its, Lisovyy, Prokhorov  and their coauthors have developed a new method to evaluate the asymptotics of the tau-functions, including the constant terms; see  <cit.>. A crucial observation is that
 the Hamiltonian is equal to the classical action differential up to a total differential.  A nice property of the action differential is that the derivative of it, with respect  to the parameters of the equation or the Stokes multipliers,  is again a total differential.
 
 Thus, the evaluation of the Hamiltonian integral is eventually reduced to the calculation of an action integral for special parameters and Stokes multipliers.
 For certain parameters and Stokes multipliers, the Hamiltonian may admit
 solutions in terms of classical special functions, and its integral can be evaluated.  Quite recently, the asymptotics of the  tau-functions for classical Painlevé equations are successfully  obtained by applying this method; see <cit.>. More general Painlevé-type  equations are also considered. For example, in <cit.>, Xu and Dai obtained the asymptotics of the Hamiltonian integral of the coupled Painlevé II system with an application in the large gap asymptotics of the Painlevé II and Painlevé XXXIV determinants.  In the present paper, however, we find we are in a different position, to use  the approach to  evaluate   the tau-function of non-classical Painlevé equations, namely the  coupled Painlevé V system in dimension four in the list of  <cit.>.

  



 §.§ The coupled Painlevé V system

To state our main results, we need to introduce the    coupled Painlevé V system in dimension four, which is a special 4-dimensional Painlevé-type
equation; see <cit.>. It is convenient to express the  coupled Painlevé V system
in the    following Hamiltonian form

    d v_k/ds=∂ H/∂ u_k ,  d u_k/ds=-∂ H/∂ v_k,    k=1,2,

 where the Hamiltonian H=H(u_1,v_1,u_2,v_2,s;α,β) is defined by

    sH=s/2H_V(u_1,v_1,s/2; α,β) -s/2H_V( u_2,v_2,-s/2; α,β)  +u_1u_2(v_1+v_2)(v_1-1)(v_2-1) ,

and H_V is the classical Hamiltonian for the Painlevé V equation, such that

    sH_V(u,v,s; α, β)=u^2v(v-1)^2-suv-α u (v^2-1)-β u (v-1)^2;

see <cit.>.
  In terms of u_k and v_k, we write (<ref>) as  the following system of equations
 
    {[    sd u_1/ds=s/2u_1-u_1^2(v_1-1)(3v_1-1)-u_1u_2(v_2-1)(2v_1+v_2-1)+2(α+β)u_1v_1-2β u_1,;  sd u_2/ds=-s/2u_2-u_2^2(v_2-1)(3v_2-1)-u_1u_2(v_1-1)(v_1+2v_2-1)+2(α+β)u_2v_2-2β u_2 ,; sd v_1/ds=-  s/2v_1+2u_1v_1(v_1-1)^2+u_2(v_1+v_2)(v_1-1)(v_2-1)-α(v_1^2-1)-β (v_1-1)^2,;   sd v_2/ds=s /2v_2+2u_2v_2(v_2-1)^2+u_1(v_1+v_2)(v_1-1)(v_2-1)-α(v_2^2-1)-β (v_2-1)^2. ].

If the parameter β=0, we have the symmetry relations

    u_2=-u_1v_1^2     v_2=1/v_1,

as shown later in Section <ref>.
Accordingly, the  system of equations  (<ref>) is reduced to
  
    {[ sd u_1/ds=1/2su_1-2u_1^2v_1(v_1^2-1)+2α u_1v_1 ,;    sd v_1/ds=-1/2sv_1+u_1(v_1^2-1)^2-α(v_1^2-1). ].

From the second equation in  (<ref>), we see that u_1 can be expressed in terms of v_1 and v_1'. Using this expression to delete u_1 from the system, we find the second order nonlinear equation for v_1
 
    d^2v_1/ds^2-2v_1/v_1^2-1(dv_1/ds)^2+1/sdv_1/ds+v_1/2s+v_1(v_1^2+1)/4(v_1^2-1)+αv_1^2+1/2s=0.

Applying  the Mobius transformation v=v_1+1/v_1-1 to (<ref>), we obtain the Painlevé III equation
 
    d^2v/ds^2-1/v(dv/ds)^2+1/sdv/ds-1/2s((α+1/2 )v^2+(α-1/2 ))-1/16v^3+1/16v=0.


 If the parameters α=β=0,  we obtain the Painlevé V  equation from (<ref>)
 after the transformation q=v_1^2, that is,
 
    d^2q/ds^2- (1/q-1+1/2q )(d q/ds)^2+1/sd q/ds+
    q/s+q(q+1)/2(q-1)=0.

Moreover, using the symmetry (<ref>), the Hamiltonian (<ref>) is reduced to the classical Hamiltonian for the Painlevé V equation after elementary transformations,

    sH(u_1,v_1,u_2,v_2,s;0,0)=u_1^2(v_1^2-1)^2-su_1v_1=p^2q(q-1)^2-spq,

where q=v_1^2 and p=u_1/v_1.  Denote

    σ(s)=sH(u_1,v_1,u_2,v_2,s;0,0),

we get  from the Hamilton equations q'=∂ H/∂ p and p'=-∂ H/∂ q the Jimbo-Miwa-Okamoto σ-form of the Painlevé V equation

    (sσ”)^2-(σ-sσ'+4σ'^2)(σ-sσ')=0;

see <cit.>.


Quite recently, the coupled Painlevé systems have found interesting applications in random matrix theory. For example, a different coupled Painlevé V system was used in the studies  of the Bessel process   and  the distribution of the ratio between the second smallest and the smallest eigenvalue in the Laguerre unitary ensemble <cit.>. The coupled Painlevé II system was applied  in the studies of the Airy point process and the determinants of the Painlevé II and Painlevé  XXXIV kernels  <cit.>.  Moreover, the coupled Painlevé III system was involved in the Painlevé-type representation of the determinants of a Painlevé III kernel, which arose  in the studies of a singularly perturbed Laguerre unitary ensemble <cit.>.

We have the following existence and asymptotic results for the coupled Painlevé V system (<ref>). The solutions play a central role in our
integral representation of the gap probability.
    For pure imaginary parameter β and α>-1/2,  there exist solutions to the coupled Painlevé V system (<ref>)
  with the following asymptotic behavior
 
    u_1(s)=-Γ(1+α+β)Γ(1+α-β)/2π iΓ(1+2α)^2e^-π iβ(|s|/2 )^2α[1+O(s^2α+1 )+O(s)],    is→0^+,


    v_1(s)=1+O (s^2α+1 )+O(s),     is→ 0^+,


    u_2(s)=Γ(1+α+β)Γ(1+α-β)/2π iΓ(1+2α)^2e^π iβ (|s|/2 )^2α[1+O (s^2α+1 )+O(s)],    is→ 0^+,


    v_2(s)=1+O (s^2α+1 )+O(s),     is→ 0^+,


    u_1(s)=i/8s(1+O(  1/s) ),    is→+∞,


    v_1(s)=i+O(1/s),    is→+∞,


    u_2(s)=i/8s(1+O(1/s)),    is→+∞,


    v_2(s)=-i+O(1/s),    is→+∞.

Moreover, the Hamiltonian H(s;α,β)=H(u_1,v_1,u_2,v_2,s;α,β), associated with these u_k(s)  and v_k(s),   is  pole-free
  for s∈ -i(0, +∞)  and has the asymptotic behavior
 
    H(s;α,β)=Γ(1+α+β)Γ(1+α-β)cos(πβ)/iπ 2^2α+1(2α+1)Γ(1+2α)^2|s|^2α+O (s^2α+1 ),    is→ 0^+,

 and
 
    H(s;α,β)=s/16+i/2α-(α^2-β^2+1/4)1/s+O(1/s^2),    is→+∞.



 For the parameters α=1/2 and β=0,   the coupled Painlevé V system (<ref>) admits solutions in terms of classical special functions; see Section <ref> for details. In particular, we have
 
    H(u_1,v_1,u_2,v_2,s;1/2,0)=s/16+i/4I_0'(is/4)/I_0(is/4),

where I_0(s) is the modified Bessel function of order zero.  It is noted that for this special solution, the integral of the Hamiltonian can be easily evaluated.



 §.§ Gap probability and Painlevé-type formulas

 The  gap probability for the eigenvalues of
  the circular unitary ensemble  with a Fisher-Hartwig singularity can be expressed in terms of the Toeplitz determinant associated with the weight function  (<ref>).
Via deriving  the asymptotics of the Toeplitz determinant,  we give an explicit expression  for the gap probability as an integral of the Hamiltonian to  the coupled Painlevé V system in dimension four.
  For pure imaginary parameter β and α>-1/2, as n→∞ and t→ 0^+ in a way such that nt is bounded, we have the asymptotic approximation of the Toeplitz determinant associated with the weight function (<ref>)

    D_n(t)/D_n(0)=exp(∫_0^-2intH(τ;α,β)dτ+O (1/n )),

where H(s;α,β) is the Hamiltonian for the coupled Painlevé V system with the properties specified in Theorem <ref> and the error term is uniform for nt  bounded.



It then follows from (<ref>) and  Theorem <ref>  that we have the Painlevé-type representation of the gap probability for  the circular ensemble with a  Fisher-Hartwig singularity (<ref>)  as n→∞

    ({θ_j∈(t,2π-t): 1⩽ j⩽ n})=exp(∫_0^-2intH(τ;α,β)dτ+O (1/n )).

The gap probability can also be expressed
in terms of the Fredholm determinant of the confluent hypergeometric kernel, as given in
(<ref>), such that

    lim_n→∞D_n(2s/n)/D_n(0)= (I-K_s^(α,β) ),

 where K_s^(α,β) is the operator with the confluent hypergeometric kernel acting on L^2(-s,s); see <cit.>.
As a direct application of the above theorem, we obtain an explicit representation of the   determinant of the confluent hypergeometric kernel.
    For pure imaginary parameter β and α>-1/2, let K_s^(α,β) be
  the operator with the confluent hypergeometric kernel K^(α,β)(u,v) in (<ref>) acting on L^2(-s,s), s>0, we have

    (I-K_s^(α,β) )=exp(∫_0^-4isH(τ;α,β)dτ),

where H(s;α,β) is the Hamiltonian for the coupled Painlevé V system with the properties specified in Theorem <ref>.


For the special parameters α=β=0,  the confluent hypergeometric kernel is reduced to the classical sine kernel  (<ref>).  Applying Theorem <ref>, Theorem <ref>
and  Remark <ref>, we recover the following  Painlevé V representation of the determinant of the sine kernel, which was obtained
earlier by Jimbo, Miwa, Môri and Sato <cit.>.
   Let K_s^(0,0) be
  the operator with sine kernel  (<ref>) acting on L^2(-s,s), s>0, we have

    (I-K_s^(0,0) )=exp(∫_0^sσ_V(τ)/τdτ),

where σ_V(s) is the solution of the equation

    (sσ_V”)^2+4(4σ_V-4sσ_V'-σ_V'^2)(σ_V-sσ_V')=0,

characterized by the asymptotic behavior

    σ_V(s)∼ -2/πs,    s→ 0^+     σ_V(s)∼ -s^2-1/4,    s→ +∞.





In view of Remark <ref> and Theorem <ref>,  for the special parameters α=1/2 and β=0, the determinant K^(α,β)_s can be evaluated explicitly to give

    (I-K^(1/2, 0)_s )=e^- s^2/2I_0(s),

where I_0(s) is the modified Bessel function of order zero, with I_0(0)=1.  It is worth noting that for these special parameters, the confluent hypergeometric kernel is equivalent to the Bessel kernel  (see <cit.>):
 
    K^(1/2, 0)(u,v)=√(|uv|)/2 J_0(u)J_0'(v)-J_0(v)J_0'(u)/u-v,

where J_0(x) is the Bessel function of order zero.  This explicit form
(<ref>)
of the determinant   furnishes   a   starting point in our asymptotic analysis of the integral of the Hamiltonian in  later sections.




 §.§ Total integral of the Hamiltonian and large gap aysmptotics


As is noted in Remark <ref> that for certain special parameters, the integral of the Hamiltonian associated can be  evaluated explicitly.
   For general parameters, the derivation is divided into two steps. First we evaluate the derivative of the Hamiltonian integral
   by deriving and applying certain differential identities, with respect  to the parameters,  for the Hamiltonian.  Then, after taking integration   we obtain the following  total integral of the Hamiltonian.

   For pure imaginary parameter β and α>-1/2, we have the total integral of the Hamiltonian after regularization  at infinity
 
    ∫_0^cH(τ;α,β)dτ+∫_c^-i∞ [H(τ;α,β)
    -τ/16-i/2α+α^2-β^2+1/4/τ ]dτ
       =c^2/32+iα c/2-(α^2-β^2+1/4)ln|c|/4
    +ln(√(π)  G( 1/2)^2 G(1+2α)/2^2α^2G(1+α+β) G(1+α-β)),

where  ic>0 and G is   the Barnes G-function.


As an application of  the above total integral of the Hamiltonian, we derive the  large gap asymptotics of the   confluent hypergeometric-kernel determiant, including the explicitly given constant term. From the large-s asymptotic formula  (<ref>), we see that the second integral in (<ref>) is of order O(1/c)
as ic→+∞. Therefore,  let c=-4is and  substitute (<ref>) into (<ref>), we obtain the following theorem as s→+∞.

  (Deift, Krasovsky and  Vasilevska  <cit.>)  For pure imaginary parameter β and α>-1/2, let K_s^(α,β) be
  the operator with the confluent hypergeometric kernel K^(α,β)(u,v) in (<ref>) acting on L^2(-s,s), we have the asymptotics as s→ +∞

    (I-K_s^(α,β))=
    √(π) G(1/2)^2G(1+2α)/2^2α^2G(1+α+β)G(1+α-β)s^-(α^2-β^2+1/4)e^-s^2/2+2α s [1+O(1/s ) ],
where G is  the Barnes G-function.

Therefore, we have reproduced the large gap asymptotics of the determinant with a confluent hypergeometric kernel, obtained previously by Deift, Krasovsky and Vasilevska <cit.>. Particularly, we recover the large gap asymptotics of the sine-kernel determinant by taking  α=β=0 in (<ref>) and using the relation between  the Barnes G-function and the  Riemann zeta-function that ln G(1/2)=3/2ζ'(-1)-1/4lnπ+1/24ln 2.

   Let K_s^(0,0) be
  the operator with sine kernel (<ref>) acting on L^2(-s,s), we have the large gap asymptotics, as s→+∞,

    ln(I-K_s^(0,0) )=-s^2/2-ln s/4+ln2/12+3ζ'(-1)+O(1/s ),

where ζ'(s) is the derivative of the Riemann zeta-function.


Using the series expansion of σ_V(s) as s→ +∞, of which the first few terms can be found in (<ref>), Dyson was able to  obtain the asymptotics of the sine-kernel determinant of the form (<ref>) in <cit.>. While the constant factor can not be derived from the series expansion of σ_V(s), it was conjectured  to be given by the  Riemann zeta-function c_0=ln2/12+3ζ'(-1) by Dyson in the same paper <cit.>.  This conjecture has been proved rigorously by Krasovsky <cit.> and Ehrhardt <cit.> using different methods.  In the present paper, our result provides another proof of the large gap asymptotics (<ref>)
 including the  constant conjectured by Dyson.

 The rest of the paper is arranged as follows. In Section <ref>,  we formulate the model Riemann-Hilbert problem (RH problem or RHP, for short) for Ψ(ζ; s). We derive a Lax pair corresponding to the model RH problem, of which the compatibility condition is expressed as a coupled Painlevé V system.   Several differential identities of the Hamiltonian  are also established, which are crucial in the evaluation of the total integral of the Hamiltonian.  We then justify the solvability of Ψ(ζ;s) for the parameters α>-1/2, β∈ iℝ and s∈ -i(0, ∞) by proving a vanishing lemma. The solvability  implies the pole-free of the Hamiltonian  associated with the coupled Painlevé V system for is>0. The special function solutions to the coupled Painlevé system  are also derived for certain specific parameters in this section.
 In Section <ref>, we derive asymptotic approximations for the coupled Painlevé V system as is→ 0^+
by using Deift-Zhou nonlinear steepest descent method for the RH problems. While the large-s analysis will be carried out in Section <ref>. These two sections together provide a proof of Theorem <ref>.
In Section <ref>, we formulate  a RH problem for Y, corresponding to the orthogonal polynomials associated with the weight function (<ref>) on an arc, we also derive a differential identity connecting the Toeplitz determinant (<ref>) with Y.
In Section <ref>, we carry out, in full details,  the Deift-Zhou nonlinear steepest descent analysis  of the RH problem for Y.
The asymptotics of the orthogonal polynomials and the differential identity  then lead to a proof of Theorem <ref> in this section.
Section <ref>  will be devoted to the evaluation of the total integral of the Hamiltonian and the large gap asymptotic analysis of the model. Theorem <ref> is also proved  in this last section.
  





§ MODEL RH PROBLEM AND THE COUPLED PAINLEVÉ V SYSTEM

In this section, we introduce a  RH problem for Ψ(ζ;s) for later use. We derive a Lax pair for the solution to the RH problem for Ψ(ζ;s), which turns out to be  a Garnier system  in the list of  <cit.>, with three regular singularities and one irregular singularity of order two. The compatibility condition of the Lax pair is described by the  coupled Painlevé V system in dimension four.
We then prove the solvability of this model RH problem and  the pole-free of the Hamiltonian associated with the
 coupled Painlevé V system for is>0. The Hamiltonian plays a   central role in the derivation of our main results on the asymptotics for the Toeplitz determinant.
In this section,  we  will also show that for special parameters, the   coupled Painlevé system  admits special function solutions.





 §.§ A Model RH problem 


We formulate a model RH problem, which will pave the road to the steepest descent analysis in later sections. The model  RH problem for 2× 2 matrix function Ψ(ζ)=Ψ(ζ; s) is the following:

  
  * (a)  Ψ(ζ; s) is analytic in
  ζ∈ℂ∖{∪^7_j=1Σ_j}, where the oriented ζ-contours
   
    Σ_1=1+e^π i/4ℝ^+, Σ_2=-1+e^3π i/4ℝ^+,  Σ_3=-1+e^-3π i/4ℝ^+, Σ_4=e^-π i/2ℝ^+, Σ_5=1+e^-π i /4ℝ^+,

   Σ_6=(0,1) and Σ_7=(-1,0), as depicted  in Figure <ref>; see also an illustration of the regions Ω_j for j=1,2,⋯,5, each having Σ_j and Σ_j+1 as portions of its boundary.

  

  
  * (b)  Ψ(ζ;s) satisfies the jump conditions
 
    Ψ_+(ζ;s)=Ψ_-(ζ;s)
     {[     [           1           0; e^-π i(α-β)           1 ],                                                    ζ∈Σ_1,;         [          1          0; e^π i(α-β)          1 ],                                                    ζ∈Σ_2,; [            1 -e^-π i(α-β);            0            1 ],                                                    ζ∈Σ_3,;                                              e^2π iβσ_3 ,                                                    ζ∈Σ_4,;     [           1 -e^π i(α-β);           0           1 ],                                                    ζ∈Σ_5,;     [           0 -e^π i(α-β); e^-π i(α-β)           0 ],                                                    ζ∈Σ_6,; [            0 -e^-π i(α-β);   e^π i(α-β)            0 ],                                                    ζ∈Σ_7. ] .

 For later use, we denote the jump  matrix on Σ_k by J_k for k=1,2,...,7.



  * (c) As ζ→∞,  we have
  
    Ψ(ζ;s)=( I  +Ψ_1(s)/ζ+Ψ_2(s)/ζ^2+
    O(1/ζ^3) ) ζ^-βσ_3 e^1/4 sζσ_3,

  where  the branch cut of the function  ζ^β is taken along (0, -i∞) such  that ζ∈ (-π/2, 3π/2),  and σ_3 is the Pauli matrix. This and the other Pauli matrices are
 
    σ_1=[ 0 1; 1 0 ],     σ_2=
                     [  0 -i;  i  0 ]    σ_3 =[  1  0;  0 -1 ].


 
  * (d) As ζ→ 0, we have
  
    Ψ(ζ;s)=Ψ^(0)(ζ; s) ζ^ασ_3 E^(0)_j,    ζ∈Ω_j,  j=1,3,4,

  where Ψ^(0)(ζ; s) =Ψ^(0)_0(s)(I+Ψ^(0)_1(s)ζ+O(ζ^2 )) is analytic at ζ=0, and ζ^α takes the principal branch.
  The connection matrices are given below

    E^(0)_1=I,    E^(0)_4=J_6^-1,    E^(0)_3=J_6^-1J_4^-1,

where   J_k is the jump matrix for Ψ on Σ_k for k=1,2,...,7; cf. (<ref>).

 
  * (e) As ζ→ 1, we have
  
    Ψ(ζ;s)=Ψ^(1)(ζ; s)    [          1 c_1ln(ζ-1);          0          1 ]E^(1)_j,    ζ∈Ω_j,  j=1,4,5,

  where  Ψ^(1)(ζ; s) =Ψ^(1)_0(s)(I+Ψ^(1)_1(s)(ζ-1)+O ((ζ-1)^2 )) is analytic at  ζ=1,
  the logarithmic  function takes the principal branch,
    the constant c_1=-1/2π ie^π i(α-β) and the connection matrices are given as
 
    E^(1)_1=I,    E^(1)_5=J_1^-1,    E^(1)_4=J_1^-1J_5^-1,

    J_k being the jump matrix for Ψ on Σ_k for k=1,2,...,7; see (<ref>).

 
  * (f) As ζ→ -1, we have
  
    Ψ(ζ;s)=Ψ^(2)(ζ; s)    [          1 c_2ln(ζ+1);          0          1 ]E^(2)_j,    ζ∈Ω_j,  j=1,2,3,

  where Ψ^(2)(ζ; s) =Ψ^(2)_0(s)(I+Ψ^(2)_1(s)(ζ+1)+O((ζ+1)^2)) is analytic at ζ=-1.
  The branch for the logarithm   is taken so that (ζ+1)∈(0,2π).
    The constant c_2=1/2π ie^-π i(α-β) and the connection matrices are given as follow

    E^(2)_1=I,    E^(2)_2=J_2^-1,    E^(2)_3=J_2^-1J_3^-1,

where   J_k is the jump matrix for Ψ on Σ_k for k=1,2,...,7; see (<ref>).


         In the above model RH problem, the connection matrices describing  the sector-wise behaviors of Ψ near the node points ζ=0 and ±1,
are specified  in a way such that they are in compliance with  the jump conditions (<ref>) in neighborhoods of  the origin and ζ=±1.




 §.§ Lax pair and the coupled Painlevé V  system

In this section, we derive a Lax pair from the RH problem for Ψ. The Lax pair is associated with a special Garnier system,  of which the compatibility condition is expressed as the coupled Painlevé  V system in dimension four.
 We have the following  Lax pair

    d/dζΨ(ζ;s)=L(ζ;s) Ψ(ζ;s),    d/dsΨ(ζ;s)=U(ζ;s) Ψ(ζ;s),

  where
  
    L(ζ;s)=s/4σ_3+A_0(s)/ζ+A_1(s)/ζ-1+ A_2(s)/ζ+1,

  
    U(ζ;s)=1/4ζσ_3+B(s),

with the coefficients  given below
   
    A_0(s)= [           u_1(s)v_1(s)+u_2(s)v_2(s)-β -( u_1(s)v_1(s)+u_2(s)v_2(s)-α-β)y(s);  (u_1(s)v_1(s)+u_2(s)v_2(s)+α-β)/y(s)          -u_1(s)v_1(s)-u_2(s)v_2(s)+β ],

  
    A_k(s)= [       - u_k(s)v_k(s)           u_k(s)y(s); -u_k(s)v_k^2(s)/y(s)         u_k(s)v_k(s) ],     k=1,2,

and
  
    B(s)=1/s[           0  b_1(s)y(s); b_2(s)/y(s)           0 ],

            where
 
    {[             b_1(s)=-u_1(s)(v_1(s)-1)-u_2(s)(v_2(s)-1)+α+β,; b_2(s)=-u_1(s)v_1(s)(v_1(s)-1)-u_2(s)v_2(s)(v_2(s)-1)+α-β. ].

The compatibility condition of the Lax pair is expressed as the coupled Painlevé V system
 
    {[  sd u_1/ds=s/2 u_1-u_1^2(v_1-1)(3v_1-1)-u_1u_2(v_2-1)(2v_1+v_2-1)+2(α+β)u_1v_1-2β u_1,; sd u_2/ds=-s/2 u_2-u_2^2(v_2-1)(3v_2-1)-u_1u_2(v_1-1)(v_1+2v_2-1)+2(α+β)u_2v_2-2β u_2,; sd v_1/ds=-s/2 v_1+2u_1v_1(v_1-1)^2+u_2(v_1+v_2)(v_1-1)(v_2-1)-α(v_1^2-1)-β (v_1-1)^2,;  sd v_2/ds=s/2 v_2+2u_2v_2(v_2-1)^2+u_1(v_1+v_2)(v_1-1)(v_2-1)-α(v_2^2-1)-β (v_2-1)^2. ].

Let

    sH=sH(u_1,v_1,u_2,v_2,s;α,β)=-s/2(Ψ_1)_11-(α^2-β^2),

  where (Ψ_1)_11 is the 11-entry of the  coefficient Ψ_1 in the large-ζ asymptotic approximation (<ref>) of Ψ(ζ;s), then we have
  
    sH=s/2H_V( u_1,v_1,s/2; α,β) -s/2H_V(u_2,v_2,-s/2; α,β)  +u_1u_2(v_1+v_2)(v_1-1)(v_2-1),

as defined in (<ref>), where H_V( u,v,s; α, β) is the Hamiltonian for the Painlevé V equation

    sH_V(u,v,s; α, β)=u^2v(v-1)^2-suv-α u (v^2-1)-β u (v-1)^2.

The system (<ref>) is equivalent to the following Hamiltonian formulation

    d v_k/ds=∂ H/∂ u_k ,  d u_k/ds=-∂ H/∂ v_k,    k=1,2.

  

 
 Note that all the jump matrices in (<ref>) of the RH problem for Ψ(ζ;s) are independent of the variables ζ and s.
 Then d/dζΨ(ζ;s), d/dsΨ(ζ;s) and Ψ(ζ;s) satisfy the same jump conditions.
 Thus, the matrix-valued functions L(ζ;s)= d/dζΨ(ζ;s)Ψ^-1(ζ;s) and U(ζ;s)= d/dsΨ(ζ;s)Ψ^-1(ζ;s)
 are meromorphic for ζ in the complex plane with only possible isolated singularities  at ζ=0, ± 1.
 Then, it follows from the local behaviors of Ψ(ζ; s) as ζ→∞, ζ→ 0, and ζ→± 1; cf. (<ref>), (<ref>), (<ref>) and (<ref>),  that L and U are rational functions in ζ and take  the form as given in (<ref>) and (<ref>), respectively.

 Using the fact that Ψ=1, we have L= U=0 and thus all the coefficients A_k, k=0,1,2 and B are trace-zero.
 Substituting the behavior of Ψ at infinity (<ref>) into the first equation of the Lax pair (<ref>), we find after comparing the
 coefficient of 1/ζ that

    A_0+A_1+A_2=[          - β -s/2(Ψ_1)_12;  s/2(Ψ_1)_21            β ],

              where Ψ_1 is the coefficient of
             1/ζ in the large-ζ  asymptotic expansion of Ψ(ζ;s) in (<ref>).
 Moreover, combining the master equation in (<ref>) with the local behavior (<ref>) of  Ψ(ζ;s) at ζ=0, and (<ref>)-(<ref>) at ζ=± 1, we have
 
    A_0=-α^2,    A_k=0,    k=1,2.

 Thus,  the coefficients A_k, k=0,1,2, can be taken in the form appeared in (<ref>) and (<ref>).

  Similarly, substituting the behavior of Ψ at infinity (<ref>) into the second equation of the Lax pair (<ref>) and comparing    the leading-order terms
  in ζ, we find

    B=[            0 -1/2(Ψ_1)_12;  1/2(Ψ_1)_21            0 ],

              where Ψ_1 is  the coefficient  of the large-ζ asymptotic approximation of Ψ(ζ;s) in (<ref>).
 This,  together with (<ref>), implies that B is given by  (<ref>).

 The compatibility condition of the Lax pair (<ref>) is readily written   as
 
    {[        d/ds A_0=[B, A_0],; d/ds A_1=[B+1/4σ_3, A_1],; d/ds A_2=[B-1/4σ_3, A_2], ].
where the commutator [A,B]=AB-BA.
 Denote  a(s) by
 
    a(s)=(A_0)_11(s)=u_1(s)v_1(s)+u_2(s)v_2(s)-β.

Recalling the definition of b_k in (<ref>) and substituting (<ref>), (<ref>) and (<ref>) into (<ref>) yields
  
    {[                                   sd/ds ((a-α)y)=2ab_1y,;                                s d/ds((a+α)/y) =2ab_2/y,;            s d/ds (u_kv_k)=b_1u_kv_k^2+b_2u_k,    k=1,2,; s  d/ds (u_ky)=(-1)^k+1s/2u_k y+2b_1u_k v_k y,    k=1,2. ].

Then, the first two equations imply that the gauge parameter y(s) of the Lax pair satisfies the equation
 
    sy'(s)/y(s)=b_1(s)-b_2(s)=u_1(s)(v_1(s)-1)^2+u_2(s)(v_2(s)-1)^2+2β.

 This equation, together with the last two equations of (<ref>), gives us the system   (<ref>).

Substituting the large-ζ expansion   (<ref>) into the first equation of the Lax pair (<ref>), and comparing the coefficients of 1/ζ^2, we find

    s/4[σ_3Ψ_1, Ψ_1]+s/4[Ψ_2, σ_3]-β[Ψ_1, σ_3]-Ψ_1=A_1-A_2.

 Hence, combining the definition (<ref>)  with (<ref>) and   (<ref>), we obtain  (<ref>). By (<ref>),   it is readily verified  that the Hamiltonian system  (<ref>) is equivalent to the system of equations (<ref>).
This completes the proof of the proposition.  



We then derive several differential identities for later use.
 Let Ψ^(1) and Ψ^(2) be the matrix functions
in the asymptotic behaviors of Ψ(ζ;s) near ζ=1 and ζ=-1, defined respectively in (<ref>) and  (<ref>), we have

    u_k(s)v_k(s)=(-1)^ki/π e^(-1)^k+1π i(α-β)d/ds(Ψ^(k)_1(s)   )_21   k=1,2.

 Moreover,  we have
 
    y(s)= (Ψ^(0)_0 (s)  )_11/ (Ψ^(0)_0 (s) )_21
 and
 
    sd/dsln y=u_1(s)(v_1(s)-1 )^2+u_2(s)(v_2(s)-1 )^2+2β.

 Denoting

    d(s)=2α(Ψ^(0)_0(s))_11(Ψ^(0)_0(s) )_21,

we have

    sd/dsln d(s)
    =b_1(s)+b_2(s)=- u_1(s)(v_1^2(s)-1 )-u_2(s)(v_2^2(s)-1 ) +2α.



Substituting  (<ref>),  (<ref>) and  (<ref>) into the first equation of  (<ref>), we find

    A_0(s)= αΨ^(0)_0(s)σ_3{Ψ^(0)_0(s)}^-1,


    A_k(s)= c_k Ψ^(k)_0(s)σ_+{Ψ^(k)_0(s)}^-1,

where σ_+=[ 0 1; 0 0 ] and the constant c_k=(-1)^k1/2π ie^(-1)^k+1π i(α-β) for k=1,2.
Similarly, upon substitution of (<ref>),  (<ref>) and  (<ref>) into the second equation of  (<ref>),
it follows that

    d/dsΨ^(0)_0(s)=BΨ^(0)_0(s),


    d/dsΨ^(k)_0(s)= ((-1)^k+1/4σ_3+B ) Ψ^(k)_0(s),

and

    d/dsΨ^(k)_1(s)= 1/4{Ψ^(k)_0(s)}^-1σ_3 Ψ^(k)_0(s)

 for k=1,2.
 We then get from (<ref>) and (<ref>) that
   
    d/ds (Ψ^(k)_1(s) )_21=1/2c_k(A_k(s) )_11

for k=1,2.
This, together with  (<ref>), implies (<ref>).


Equation (<ref>) follows from a combination of (<ref>) and (<ref>). Indeed, we have

    y(s)=(A_0(s) )_11+α/(A_0(s) )_21= (Ψ^(0)_0 (s)  )_11/ (Ψ^(0)_0 (s) )_21.
Here use has been made of the fact that Ψ^(0)_0 (s)=1.
In view of (<ref>) and (<ref>), and splitting
  (<ref>) to entries, we derive the logarithmic derivative (<ref>)
for the gauge parameter d(s). The equation (<ref>) has already been
derived before; see (<ref>).  This completes the proof.


The following differential identities of the Hamiltonian with respect  to the parameters α and β are crucial to the evaluation of the total integral of the Hamiltonian.

For the Hamiltonian H=H(u_1,v_1,u_2,v_2,s;α,β) it holds

    H=(u_1dv_1/ds+u_2dv_2/ds-H)+d/ds(sH+αln d(s)-βln y(s)-2(α^2-β^2)ln s),

where y(s) and d(s) satisfy the properties given in Proposition <ref>.
The Hamiltonian system implies the following useful differential formulas

    d/dα(u_1dv_1/ds+u_2dv_2/ds-H)=d/ds(u_1d/dαv_1+u_2d/dαv_2-ln d+2αln s ),


    d/dβ(u_1dv_1/ds+u_2dv_2/ds-H)=d/ds (u_1d/dβv_1+u_2d/dβv_2+ln y-2βln s ).



It follows from the Hamiltonian system (<ref>)  and (<ref>)  that

    d/ds(sH(s))=-1/2(u_1(s)v_1(s)-u_2(s)v_2(s)).

Using this equation, together with  (<ref>), (<ref>) and (<ref>),  we readily verify (<ref>).
The verification of (<ref>) is straightforward, by substituting
(<ref>),  (<ref>) and (<ref>) into it.
While that of (<ref>)   follows from (<ref>),  (<ref>)  and (<ref>). Here we regard u_k and v_k, k=1,2, as functions of α and β, and α, β and s are independent.
 



 

 §.§ Vanishing lemma and existence of solution to the model RH problem  

We will prove the existence of solution to the model RH problem for Ψ(ζ;s) if the parameters  α>-1/2, β∈ i ℝ and s∈ -i(0, +∞).  We start with the vanishing lemma which shows that the homogeneous RH problem has only zero solution.
 
 For α>-1/2, β∈ i ℝ and s∈ -i(0, +∞), we suppose that Ψ̂(ζ) satisfies the same jump conditions (<ref>) and the same  behaviors (<ref>), (<ref>) and (<ref>) as Ψ(ζ), respectively as ζ tends to the origin and ±1. Further assume the behavior of Ψ̂(ζ) to be
  
    Ψ̂(ζ)=
    O(1/ζ)ζ^-βσ_3 e^1/4 sζσ_3,   ζ→∞.

   Then, we have Ψ̂(ζ)=0 for ζ∈ℂ.
 

 
 To normalize the behavior as ζ→∞, we introduce the first transformation
 
    Ψ̂^(1)(ζ)=Ψ̂(ζ)e^-1/2π iβσ_3φ(ζ)^βσ_3e^-1/4 sζσ_3,  ζ∉Σ_4∪Σ_6∪Σ_7,

   where the contours are illustrated in  Figure <ref>, and
   
    φ(ζ)=ζ+√(ζ^2-1).

 The branch for √(ζ^2-1) is taken such that  (ζ± 1)∈(-π, π) and the branch for ζ^β is  taken as in (<ref>), such that ζ∈(-π/2, 3π/2).
 Then, we have the normalized  behavior at infinity
  
    Ψ̂^(1)(ζ)=
    O(1/ζ),

and the modified jump conditions
 
    Ψ̂^(1)_+(ζ)=Ψ̂^(1)_-(ζ)
     {[                         [                      1                      0; e^-π iαφ(ζ)^2βe^-1/2sζ                      1 ],                                                                                                                    ζ∈Σ_1,;         [                          1                          0; e^π i(α-2β)φ(ζ)^2βe^-1/2sζ                          1 ],                                                                                                                    ζ∈Σ_2,; [                            1 -e^-π i(α-2β)φ(ζ)^-2βe^1/2sζ;                            0                            1 ],                                                                                                                    ζ∈Σ_3,;                         [                      1 -e^π iαφ(ζ)^-2βe^1/2sζ;                      0                      1 ],                                                                                                                    ζ∈Σ_5,;                                                 [                0   -e^π iαe^1/2sζ; e^-π iα e^-1/2sζ                0 ],                                                                                                                    ζ∈Σ_6,;                                                 [                0 -e^-π iα e^1/2sζ;   e^π iαe^-1/2sζ                0 ],                                                                                                                    ζ∈Σ_7. ] .



 To deform the jump contours to the real axis, we take the second transformation
 
    Ψ̂^(2)(ζ)=Ψ̂^(1)(ζ)
     {[                     [                      1                      0; e^-π iαφ(ζ)^2βe^-1/2sζ                      1 ],                                                                                                           ζ>0, ζ∈Ω_5,;                         [                     1 e^π iαφ(ζ)^-2βe^1/2sζ;                     0                     1 ],                                                                                                           ζ<0, ζ∈Ω_5,;     [                          1                          0; e^π i(α-2β)φ(ζ)^2βe^-1/2sζ                          1 ],                                                                                                           ζ>0, ζ∈Ω_2,; [                           1 e^-π i(α-2β)φ(ζ)^-2βe^1/2sζ;                           0                           1 ],                                                                                                           ζ<0, ζ∈Ω_2,;                                                                                                                    I,                                                                                                                     . ] .

Then, Ψ̂^(2)(ζ) is analytic in ℂ∖ℝ, with jump condition   on the
real axis as follows:
 
    Ψ̂^(2)_+(ζ)=Ψ̂^(2)_-(ζ)Ĵ(ζ),     ζ∈ℝ,

where the orientation is from left to right, and the jump matrices are

    Ĵ(ζ)= {[                         [                      0 -e^π iαφ(ζ)^-2βe^1/2sζ; e^-π iαφ(ζ)^2βe^-1/2sζ                      1 ],                                                                                                                     ζ >1,;                                                     [               0  -e^π iαe^1/2sζ; e^-π iαe^-1/2sζ               0 ],                                                                                                                  0< ζ <1,;                                                     [               0 -e^-π iαe^1/2sζ;  e^π iαe^-1/2sζ               0 ],                                                                                                                  -1<ζ <0,; [                            0 -e^-π i(α-2β)φ(ζ)^-2βe^1/2sζ;   e^π i(α-2β)φ(ζ)^2βe^-1/2sζ                            1 ],                                                                                                                     ζ<-1.;                                                                                                                           ] .

Moreover, Ψ̂^(2)(ζ) has at most logarithmic singularities at ± 1 and fulfills  the asymptotic estimates
 
    Ψ̂^(2)(ζ)=O(1)ζ^ασ_3    ζ∈Ω_1, ζ→ 0,

and
 
    Ψ̂^(2)(ζ)=
    O(1/ζ)     ζ→∞.


 Next, we consider
   
    Q(ζ)= Ψ̂^(2)(ζ)( Ψ̂^(2)(ζ̅))^*, ζ∈ℂ∖ℝ,

where ( Ψ̂^(2)(ζ))^* denotes the Hermitian conjugate of Ψ̂^(2)(ζ).
We see that Q(ζ) is analytic for ζ∈ℂ∖ℝ. For α>-1/2, we have

    ∫_ℝQ_+(x)dx= ∫_ℝΨ̂^(2)_+(x)(Ĵ^-1(x))^*( Ψ̂^(2)_+(x))^* dx=0,

where the jump Ĵ(x) is  introduced in (<ref>).
 Adding to (<ref>) its Hermitian conjugate, for purely imaginary s and β, we have
 
    ∫_-∞^-1Ψ̂^(2)_+(x) [ 1 0; 0 0 ]( Ψ̂^(2)_+(x))^* dx+∫^∞_1Ψ̂^(2)_+(x) [ 1 0; 0 0 ]( Ψ̂^(2)_+(x))^* dx=0.

  Thus, the first column of Ψ̂^(2)_+(x)  vanishes for x>1 and x<-1.  From (<ref>) one sees that  Ψ̂^(2)(ζ) can
  be analytically extended  from the upper-half   complex plane to the lower-half, acrossing (1,∞). Thus,  the first column of Ψ̂^(2)(ζ) vanishes in the upper-half ζ-plane. This and the jump condition (<ref>) then imply the vanishing of the second column of Ψ̂^(2)(ζ) in the lower-half   plane.

To show the vanishing of the remaining column of   Ψ̂^(2)(ζ) in the whole plane, we consider two
scalar functions defined as

    g_k(ζ) ={[ -(Ψ̂^(2)(ζ))_k2e^-sζ/4φ(ζ)^2βe^-π iα,                                  ζ>0,;                 (Ψ̂^(2)(ζ))_k1e^sζ/4,                                  ζ<0, ].

for k=1,2.
Then,  it follows from (<ref>) and (<ref>)  that g_k(ζ) is analytic and bounded
for ζ∉(-∞,1] and g_k(ζ)=O(e^-|sζ|/4) for purely imaginary ζ
 and is>0. Applying Carlson's theorem,
we have g_k(ζ)=0 for k=1,2. This then implies that the second column of   Ψ̂^(2)(ζ) vanishes in the upper-half ζ-plane, and the first column of   Ψ̂^(2)(ζ) also vanishes in the lower-half  plane. This completes the proof the lemma.
   

By a standard analysis <cit.>, the vanishing lemma implies the existence of  unique solution to the RH problem for Ψ(ζ;s) for
the parameters α>-1/2, β∈ i ℝ and s∈ -i(0, +∞).


 For α>-1/2, β∈ i ℝ and s∈ -i(0, +∞), there exists unique solution to the RH problem for Ψ(ζ;s).
Then, Ψ(ζ;s) is a global function of s on -i(0,+∞), that is, it is free of poles  for s∈ -i(0, +∞). Particularly, the Hamiltonian H(s)  (<ref>) is free of poles for s∈ -i(0, +∞).
 


 

 

 §.§ Special function solutions

Now we   show that the coupled Painlevé V system (<ref>) admits special function solutions for some particular parameters.
The special solutions are useful in our derivation of the total integral of the Hamiltonian.

For β=0, the jump on Σ_4 vanishes; cf. (<ref>). It is readily verified  that σ_1Ψ(-ζ)σ_1 also solves  the RH problem for Ψ(ζ), formulated in Section <ref>.
 Then, it follows from the uniqueness of solution to the RH problem for Ψ(ζ) that

    σ_1Ψ(-ζ)σ_1=Ψ(ζ),
where σ_1 is the Pauli matrix; cf. (<ref>).
Substituting   (<ref>) into (<ref>),  we have the constraint
Ψ^(0)_0(s)=σ_1 Ψ^(0)_0(s)σ_3. Noting that Ψ^(0)_0(s)=1, we can write

    Ψ^(0)_0(s)= I-iσ_2 /√(2)l(s)^σ_3

for a certain scalar  function l(s).
We define

    P(ζ)=e^-1/4π iσ_3ζ^-1/4σ_3(I+iσ_2)/√(2)Ψ(√(ζ))e^1/4π iσ_3{[                  I,              0<ζ<π,; ([  0 -1;  1  0 ]),             -π<ζ<0. ].

Then P(ζ)  satisfies the following model RH problem.





  
  * (a)   P(ζ) is analytic in
  ℂ\{⋃_k=1^4Σ_P,k}, where
  Σ_P,1=1+e^π i/3ℝ^+, Σ_P,2=(0,1),
  Σ_P,3=1+e^-π i /3ℝ^+ and  Σ_P,4=(1,+∞), as illustrated in Figure <ref>.
  
  * (b)   P(ζ)  satisfies the jump condition
  
    P_+(ζ)=P_-(ζ) J_P,k(ζ),   ζ∈Σ_P,k,    k=1,2,3,4,

where

    J_P,1=([             1             0; e^-π i(α-1/2)             1 ]),   J_P,2=e^π i(1/2-α )σ_3,  J_P,3=([            1            0; e^π i(α-1/2)            1 ]),  J_P,4=iσ_2.


  
  * (c)  P(ζ) has the asymptotic behavior as ζ→∞
  
    P(ζ)=ζ^-1/4σ_3I-iσ_1/√(2) (I+O (1/√(ζ) ))e^1/4s√(ζ)σ_3,    0<ζ<π ,

and
 
    P(ζ)   =ζ^-1/4σ_3I-iσ_1/√(2) (I+O (1/√(ζ) ))e^1/4s√(ζ)σ_3(-iσ_2) 
       =ζ^-1/4σ_3e^-π i/2σ_3I-iσ_1/√(2) (I+O (1/√(ζ) ))e^-1/4s√(ζ)σ_3,
          -π<ζ<0.


  
  * (d)  P(ζ) satisfies the asymptotic behavior  near the origin
  
    P(ζ)=(I+c(s)σ_+)(I+O(√(ζ)))l(s)^σ_3ζ^(α/2-1/4)σ_3,
where c(s) is a scalar function depending only on s, l(s) is introduced in (<ref>), and the branch of the power function is chosen such that ζ∈ (0, 2π).
  
  * (e)  P(ζ) satisfies the asymptotic behavior  as ζ→ 1
  
    P(ζ)=O(ln(ζ-1)).




The RH problem is equivalent to the model RH problem considered by the authors in <cit.>. We can write 
    P(ζ;s)=ie^-π i/4σ_32^-1/2σ_3σ_3σ_1Ψ_0^XZ (1-ζ/4,is )σ_3,

where Ψ_0^XZ(ζ,s) is the solution to the model problem for Ψ_0(ζ,s), formulated  in <cit.>, with parameters γ=-1/2 and Θ=1/2-α  therein.

If we further assume that the parameter α= 1/2, the RH problem for P(ζ) can be solved explicitly in terms of the modified Bessel functions

    P(ζ)=(I-I_0'(|s|/4)/I_0(|s|/4)σ_-)e^-π i/4σ_3(|s|/4)^1/2σ_3σ_3Φ_B(s^2/16(ζ-1))σ_3,

where Φ_B(ζ) is defined in (<ref>), and σ_-=[ 0 0; 1 0 ].  The  lower triangular matrix is chosen to ensure the fulfilment of the asymptotic behavior (<ref>).   From (<ref>), we have
the large-ζ behavior

    [                                                                                                                                                                           P(ζ)= ζ^-1/4σ_3I-iσ_1/√(2){I+1
       /√(ζ)(1/2s(
                         [ -1 2i; 2i  1;    ])-s/8σ_3-I_0'(|s|/4)/2I_0(|s|/4)(
                         [   i   1;   1 - i;     ])) +O (1/ζ ) };                                                                                                                                                                                                                                                                                                                                                 × e^1/4s√(ζ)σ_3 ]

for 0<ζ<π.
Moreover, it follows from (<ref>)  that P(ζ) satisfies the differential equation

    dP(ζ)/dζ=(([    0    0; s/8i    0;      ])+is/8(ζ-1)(  [      I_0'(|s|/4)/I_0(|s|/4)                           1; -I_0'(|s|/4)^2/I_0(|s|/4)^2     -I_0'(|s|/4)/I_0(|s|/4);                             ]))P(ζ).

From the symmetry of Ψ(ζ)  in (<ref>) for β=0, we have
 
    u_1v_1=-u_2v_2,    u_2=-u_1v_1^2,    v_2=1/v_1,    y(s)=1.

Substituting (<ref>) into (<ref>), we see that P(ζ) solves  the differential equation with coefficients involving  u_1(s) and v_1(s)

    dP(ζ)/dζ=(([    0    0; s/8i    0;      ])+u_1/2(ζ-1)(  [      1-v_1^2  -i(v_1+1)^2; - i(v_1-1)^2      v_1^2-1;              ]))P(ζ).


Now we are in a position   to derive the special function solutions to the  coupled Painlevé system of equations (<ref>) for α=1/2 and β=0.
The integral of the Hamiltonian associated with the special function solutions is explicitly calculated, and furnishes as  boundary condition in  our evaluation of the total integral of the Hamiltonian for general parameters later.


For α=1/2 and β=0, the coupled Painlevé system of equations (<ref>) admits the following special function solutions

    u_1(s)=s/16(i+I_0'(|s|/4)/I_0(|s|/4))^2,    u_2(s)=-s/16(i-I_0'(|s|/4)/I_0(|s|/4))^2,


    v_1(s)=i-I_0'(|s|/4)/I_0(|s|/4)/i+I_0'(|s|/4)/I_0(|s|/4),    v_2(s)=i+I_0'(|s|/4)/I_0(|s|/4)/i-I_0'(|s|/4)/I_0(|s|/4),

and the Hamiltonian associated with u_k(s) and v_k(s) takes the form

    H(u_1,v_1,u_2,v_2,s;  1/2,0)=s/16+i/4I_0'(|s|/4)/I_0(|s|/4),

where

    I_0(s)=∑_n=0^∞1/n!^2(s^2 /4)^n

is the modified Bessel function of order zero.


In this specific case when
  α=1/2 and β=0,  the special function solutions  u_1(s) and v_1(s)  in (<ref>) and (<ref>) are readily derived by comparing (<ref>) with (<ref>). We further obtain   u_2(s) and v_2(s)  in (<ref>) and (<ref>)
  by using the relation (<ref>). The special Hamiltonian of the form (<ref>) follows from (<ref>), (<ref>), (<ref>) and (<ref>).    This completes the proof of the proposition.
  

  

  

§ ASYMPTOTICS OF THE COUPLED PAINLEVÉ V SYSTEM AS IS→ 0^+

In this section, we  derive  the  asymptotic approximations  of the solutions to the  coupled Painlevé V system as is→ 0^+ by performing Deift-Zhou nonlinear steepest descent analysis  <cit.> of  the RH problem  for Ψ(ζ;s).



 §.§ Outer parametrix


Let
 
    X(ξ; s)=(|s|/2)^-βσ_3Ψ(2ξ /|s|; s).

  Then, X(ξ) satisfies a re-scaled version of the RH problem for Ψ(ζ) in the complex ξ-plane with ξ=|s|ζ/2. Note that,
  the jump contour (-1,1) in the ζ-plane is mapped onto  (-|s|/2, |s|/2) in the ξ-plane. As is → 0^+,  we consider the
  following approximate RH problem Φ(ξ) by ignoring the jump condition  for X(ξ) along the shrinking line segment (-|s|/2, |s|/2).





  §.§.§ RH problem  for Φ


  
  * (a)  Φ(ξ) is analytic in
  ℂ∖{∪^5_j=1Σ_Φ,j}, where the oriented contours
         
    Σ_Φ,1=e^π   i/4ℝ^+,   Σ_Φ,2=e^3π i/4ℝ^+,   Σ_Φ,3=e^-3π i/4ℝ^+,  Σ_Φ,4=e^-π i/2ℝ^+,  Σ_Φ,5=e^-π i  /4ℝ^+,

as illustrated in Figure <ref>.
  

  
  * (b)  Φ(ξ) satisfies the jump conditions
 
    Φ_+(ξ)=Φ_-(ξ)
     {[                                       [           1           0; e^-π i(α-β)           1 ],                                                                                    ξ∈Σ_Φ,1,;                                           [          1          0; e^π i(α-β)          1 ],                                                                                    ξ∈Σ_Φ,2,;                                   [            1 -e^-π i(α-β);            0            1 ],                                                                                    ξ∈Σ_Φ,3,;                                                                                e^2π iβσ_3 ,                                                                                    ξ∈Σ_Φ,4,; [.2cm]
                                   [           1 -e^π i(α-β);           0           1 ],                                                                                    ξ∈Σ_Φ,5. ] .




  * (c) As ξ→∞,  we have
  
    Φ(ξ)=( I  +Φ_1/ξ+
    O(1/ξ^2) ) ξ^-βσ_3 e^-i/2ξσ_3,

 where Φ_1=  [            - ( α^2-β^2)i -e^-π iβΓ(1+α-β)/iΓ(α+β);   e^π iβΓ(1+α+β)/iΓ(α-β)              (α^2-β^2 )i ] and
 ξ∈ (-π/2, 3π/2).


 
  * (d) The description of the asymptotic behavior  of Φ(ξ) as ξ→ 0  is divided into two cases, namely
 2α∉ℕ and   2α∈ℕ.

  In the first case, if 2α∉ℕ,  there exists a  function Φ^(0)(ξ), analytic at ξ=0,  such that
  
    Φ(ξ)= Φ^(0)(ξ)ξ^ασ_3 C_j,   ξ→ 0, ξ∈Ω_Φ, j,  j=1,2,3,4,5,

  where Ω_Φ, j are the sectors illustrated in Figure <ref>, the branch of ξ^ασ_3 is chosen such that
  ξ∈ (-π/2, 3π/2),
  the constant matrix
  
    C_1= [                     1 sin(π(α+β))/sin (2πα);                     0                     1 ],     2α∉ℕ,  α>1/2,

and the other constant matrices are determined by C_1 and the jump conditions.


In the second case, if 2α∈ℕ,  there exists a  function Φ^(0)(ξ), analytic at ξ=0,  such that
  
    Φ(ξ)= Φ^(0)(ξ)ξ^ασ_3[                       1 (-1)^2αsin(π(α+β))/πlnξ;                       0                       1 ]Ĉ_j,   ξ→ 0, ξ∈Ω_Φ, j,  j=1,2,3,4,5,

  where Ω_Φ, j are the sectors illustrated in Figure <ref>, the branches of ξ^ασ_3 and lnξ  are chosen such that
  ξ∈ (-π/2, 3π/2),
  the constant matrix Ĉ_1 is the identity matrix and the other constant matrices are determined by Ĉ_1 and the jump conditions.


The connection matrices C_j are specified so that the asymptotic behavior of Φ near the origin be in accordance with  the jump conditions, that is,
for ξ∈Σ_Φ,4, we have the constraint

    Φ_-(ξ)^-1Φ_+(ξ)=J_3J_2 C_1^-1e^-2π iασ_3C_1J_1^-1J_5^-1=e^2π iβσ_3,
where J_j denote  the jumps on Σ_Φ,j, given in (<ref>). These are the same as the jumps on Σ_j; see (<ref>). Similarly, we readily verify that the jump conditions are also fulfilled by (<ref>) in the case 2α∈ℕ.




The solution to the above RH problem can be constructed explicitly in terms of the confluent hypergeometric function; see <cit.> and <cit.>.  The parametrix is equivalent to the one, namely M(ξ), used in  <cit.> after some elementary transfromation
 
    Φ(ξ)=e^π i β/2σ_3σ_3M(e^π i/2ξ)σ_3,   ξ∈ (-π/2,3π/2).

The analytic function Φ^(0)(ξ) brought in (<ref>) now takes

    Φ^(0)(ξ)= e^-iξ/2[   e^-π i(α+β)/2Γ(1+α-β)/Γ(1+2α)ψ(α+β,1+2α,iξ)     - e^π i(α-β)/2Γ(2α)/Γ(α+β)ψ(-α+β,1-2α,iξ); e^-π i(α-β)/2Γ(1+α+β)/Γ(1+2α)ψ(1+α+β,1+2α,iξ)      e^π i(α+β)/2Γ(2α)/Γ(α-β)ψ(1-α+β,1-2α,iξ) ],

where ψ is the confluent hypergeometric function given in (<ref>).
  Also,  the first column of Φ̂^(0)(ξ) in (<ref>) is the same as that of Φ^(0)(ξ); see <cit.>.

It is straightforward to see that Φ(ξ) shares  the same jump conditions with X(ξ) for |ξ|>|s|/2. For some small constant radius ϵ>0,  we intend to  construct a local parametrix L(ξ) in a neighborhood U(0;ϵ) of the origin,  which shares the same jump conditions as X(ξ) and
  matches with Φ(ξ) on the boundary of  U(0;ϵ), namely, on |ξ|=ϵ.



 §.§ Local parametrix


      

  §.§.§ RH problem for L


  
  * (a)  L(ξ) is analytic in U(0;ϵ ) ∖Σ^X, where Σ^X denotes the jump contours for X(ξ); see Figure
      <ref> for a re-scaled version.



  
  * (b) L(ξ)  satisfies the same jump conditions  on U(0;ϵ ) ∩Σ^X  as X(ξ).
   
  * (c)  As is→ 0^+, we have the matching condition
     
    L(ξ)=(I+O(s^2α+1))Φ(ξ),

for |ξ|=ϵ with  a constant ϵ>0.



From the asymptotic behavior
(<ref>) and (<ref>) of Φ(ξ) near the origin, we may  seek a solution to the above RH problem of the form

    L(ξ)= Φ^(0)(ξ)  [              1 |s|^2αk(ξ/|s|);              0              1 ]ξ^ασ_3 C_j       2α∉ℕ,  α>-1/2,

or
  
    L(ξ)= Φ^(0)(ξ)  [              1 |s|^2αk(ξ/|s|);              0              1 ]ξ^ασ_3[                       1 (-1)^2αsin(π(α+β))/πlnξ;                       0                       1 ]Ĉ_j       2α∈ℕ,

                                 for ξ/|s| ∈Ω_j,  j=1,2,3,4,5; cf. Figure
                                 <ref>,
   where  Φ^(0)(ξ) and C_j are given in (<ref>) and (<ref>),
   Φ^(0)(ξ) and Ĉ_j are given in (<ref>).
   The branches of ξ^α and lnξ are chosen so that ξ∈(-π/2, 3π/2). The scalar function k(ξ), to be determined, is analytic  in ℂ∖ [-1/2,1/2].

   In view of  (<ref>) and (<ref>), we see that L(ξ) has the  same jumps as X(ξ) on  Σ^X∖ [-|s|/2,|s|/2].
To  ensure the fulfilment by L(ξ)  of the jump condition  on  the interval (-|s|/2,|s|/2) as X(ξ), and the matching condition (<ref>), we look for the function k(ξ), as  a solution of the following scalar RH problem for α>-1/2.

   

  §.§.§ RH problem for k


  
  * (a)  k(ξ) is analytic in ℂ∖ [-1/2,1/2].

  
  * (b) k(ξ)  satisfies the same jump relations
   
    k_+(x)-k_-(x)={[   -e^π i(α-β)x^2α,       x∈ (0, 1/2),; -e^π i(α+β)|x|^2α,      x∈ (-1/2, 0). ].


  * (c)  As ξ→∞,  k(ξ)=O(1/ξ).
 
 Applying the Sokhotski-Plemelj formula, we obtain an explicit representation
 
    k(ξ)=-e^π i(α-β)/2π i∫_0^1/2x^2α dx/x-ξ
    -e^π i(α+β)/2π i∫_-1/2^0  |x|^2α dx/x-ξ    ξ∈ℂ∖ [-1/2,1/2].

Having had k(ξ) as in (<ref>), we readily verify that L(ξ) in (<ref>)
shares the same jumps as   X(ξ)  on real intervals. Indeed,
for ξ∈(0,|s|/2), we have

    L_-(ξ)^-1L_+(ξ)=J_5J_1(I-e^π i(α-β)σ_+)=J_6,

where J_k denotes the constant jump of Ψ(ζ) on the contour Σ_k; cf. (<ref>) and (<ref>).
Now we turn to ξ∈(-|s|/2,0).
First, with the branch  chosen such that ξ∈(-π/2,3π/2), we have

    ξ_+^α=ξ_-^α=|ξ|^α e^π iα    (lnξ)_+ =(lnξ )_-
for ξ on the negative real axis.
Therefore,  we have

    L_-(ξ)^-1L_+(ξ)=J_3J_2(I-e^-π i(α-β)σ_+)=J_7.

Thus,  L(ξ) has the same jump   as X(ξ) for ξ∈(-|s|/2,0)∪(0, |s|/2), and hence for   ξ∈ U(0;ϵ ) ∩Σ^X.
Accordingly,  we  see that X(ξ)L(ξ)^-1  is analytic in 0< |ξ|<ϵ.

To show the  analyticity of  X(ξ)L(ξ)^-1 at ξ=0,  we study the behavior of L(ξ) near the origin. From (<ref>), we have
for 2α∉ℕ

    k(ξ)=k_r(ξ)+{[        -sin ( π(α+β))/sin (2πα)ξ^2α,                             ξ∈(0,π),; sin ( π(α-β))/sin (2πα)(ξ e^π i)^2α,                           ξ∈(-π, 0), ] .

where k_r(ξ) is an analytic function in   |ξ|<1/2.
Similarly, for 2α∈ℕ, we obtain from  (<ref>) that

    k(ξ)=
    k̂_r(ξ)+{[                (-1)^2α+1sin(π(α+β))/πξ^2αlnξ,                                      ξ∈(0,π),; (-1)^2α+1sin(π(α+β))/πξ^2αlnξ+e^π i(α-β)ξ^2α,                                    ξ∈(-π, 0), ] .

where k̂_r(ξ) is an analytic function in   |ξ|<1/2.

For 2α∉ℕ and α>-1/2, a combination of (<ref>), (<ref>), (<ref>), (<ref>) and
(<ref>) shows that X(ξ)L(ξ)^-1 is  bounded in a neighborhood of   ξ=0.  While for 2α∈ℕ, it follows from (<ref>), (<ref>),
(<ref>), (<ref>) and
(<ref>)
that  X(ξ)L(ξ)^-1 is also bounded
in a neighborhood of   ξ=0.
 Thus, the singularity at the   origin is  removable. Therefore, X(ξ)L(ξ)^-1 is  analytic for |ξ|<ϵ.

For later use, we  also derive from (<ref>) that

    k(ξ)∼e^π i(α+β)/2π i2^-2αln(ξ+1/2)       ξ→ -1/2,

and

    k(ξ)∼- e^π i(α-β)/2π i2^-2αln(ξ-1/2)       ξ→1/2.



 §.§ Final transformation

Next, we define

    Z(ξ)={[ X(ξ)Φ(ξ)^-1,       |ξ|>ϵ,; X(ξ)L(ξ)^-1,       |ξ|<ϵ. ].

As shown earlier in this section,   Z(ξ) is analytic for |ξ|≠ϵ.  From (<ref>), (<ref>), (<ref>)  and (<ref>), we have  the       estimate for the jump of Z(ξ)

    J_Z(ξ)=I+c_α,β|s|^2α+1/ξΦ^(0)(ξ)σ_+Φ^(0)(ξ)^-1+
    O(s^2α+2)      is→ 0^+,

where the constant c_α,β= e^π iαcos(βπ) /iπ 2^2α+1(2α+1), and  the error term is uniform for ξ on the   clockwise jump contour  |ξ|=ϵ. Therefore, the matching condition (<ref>) is fulfilled. Hence, the RH problem for Z(ξ) is a small-norm RH problem.
By a standard analysis  <cit.>, we have the piecewise   approximation as is→ 0^+

    Z(ξ)={[                              I+c_α,β|s|^2α+1/ξΦ^(0)(0)σ_+{Φ^(0)(0)}^-1+
                        ε_Z(ξ),                                                                                             |ξ|>ϵ ,; I+c_α,β|s|^2α+1/ξ[Φ^(0)(0)σ_+{Φ^(0)(0)}^-1-
                         Φ^(0)(ξ)σ_+{Φ^(0)(ξ)}^-1 ]+ε_Z(ξ),                                                                                            |ξ|<ϵ  , ].

 where the error term ε_Z(ξ)= O(s^2α+2) as  is→ 0^+, uniformly for all ξ in the complex plane.

 



 §.§ Proof of Theorem <ref>: Asymptotics of sH as is→ 0^+

Now we have the following asymptotic approximation of X(ξ)=X(ξ;s), introduced in (<ref>):

    X(ξ)={[ Z(ξ)Φ(ξ),    |ξ|>ϵ,; Z(ξ)L(ξ),    |ξ|<ϵ; ].
cf. (<ref>).
Denoting
 
    X(ξ)=I+X_1(s)/ξ+O(1/ξ^2)     ξ→∞,

 and taking  (<ref>) and  (<ref>) into account, we have
 
    (Ψ_1)_11=2(X_1)_11/|s|.

 Using this together with (<ref>), (<ref>), (<ref>) and (<ref>), we find
 
    H   =-(Ψ_1)_11/2-(α^2-β^2)/s
        =-1/is((Z_1)_11+(Φ_1)_11)-(α^2-β^2)/s
       =Γ(1+α+β)Γ(1+α-β)cos(πβ)/iπ 2^2α+1(2α+1)Γ(1+2α)^2|s|^2α+O(s^2α+1).
This gives (<ref>).

Next, we derive the small-s behavior of y(s), d(s), and other functions. It follows from (<ref>),  (<ref>) and (<ref>) that
 
    Ψ(2ξ/|s|)=(|s|/2)^βσ_3X(ξ)=(|s|/2)^βσ_3 [I+O (s^2α+1 ) ] L(ξ)       is→ 0^+

 for |ξ|<ϵ.
Substituting  (<ref>) and  (<ref>) into (<ref>), and using the relations
(<ref>) and (<ref>),  we obtain

    y(s)=Γ(1+α-β)/Γ(1+α+β)e^-π iβ(|s|/2 )^2β[1+O(s^2α+1 )],    is→0^+,

and

    d(s)=2αΓ(1+α-β)Γ(1+α+β)/Γ(1+2α)^2e^-π iα(|s|/2 )^2α[1+O(s^2α+1)],     is→ 0^+.

Similarly, the relation (<ref>), together with (<ref>),   (<ref>),  (<ref>) and  (<ref>), gives us

    u_1(s)=-Γ(1+α+β)Γ(1+α-β)/2π iΓ(1+2α)^2e^-π iβ(|s|/2 )^2α[1+O(s^2α+1)+O(s)],    is→0^+,


    v_1(s)=1+O(s^2α+1)+O(s),     is→0^+,


    u_2(s)=Γ(1+α+β)Γ(1+α-β)/2π iΓ(1+2α)^2e^π iβ(|s|/2)^2α[1+O(s^2α+1)+O(s)],    is→0^+,

and

    v_2(s)=1+O(s^2α+1)+O(s),     is→0^+.

We mention that the error term O(s) in the above formulas  has its roots back  in the approximation of Φ^(0)(ξ) in (<ref>) as
is→ 0^+

    Φ^(0)(± |s|/2)=Φ^(0)(0)+O(s).

These are formulas (<ref>), (<ref>), (<ref>) and (<ref>), respectively. Thus, we have proved the small-s part of Theorem <ref>.


For later use, comparing (<ref>) and (<ref>) with (<ref>),   (<ref>),  (<ref>) and  (<ref>),
we  also derive the error order estimates

    (Ψ^(1)_1(s))_21=O(s^2α+1)+O(s),    (Ψ^(2)_1(s))_21=O(s^2α+1)+O(s),     is→0^+.


 
 

§ ASYMPTOTICS OF THE COUPLED PAINLEVÉ V SYSTEM AS IS→ +∞

In this section, we  derive  the  asymptotics of the    coupled Painlevé V system as is→ +∞ by using Deift-Zhou nonlinear steepest descent analysis of the  RH problem for Ψ(ζ;s).


 §.§ Normalization

We introduce a normalization of Ψ(ζ;s) at infinity of the form

    A(ζ)=Ψ(ζ;s)exp(-sg(ζ)σ_3),

 where
 
    g(ζ)=1/4√(ζ^2-1)
is analytic in ζ∈ℂ∖[-1,1], such that
  (ζ± 1)∈(-π,π).


  §.§.§  RH problem  for A


  
  * (a)  A(ζ) is analytic in
  ℂ∖{∪^7_j=1Σ_j}, where the contours  are shown  in Figure <ref>.
 
  * (b)  A(ζ) satisfies the jump conditions
  
    A_+(ζ)=A_-(ζ)
     {[     [                    1                    0; e^-π i(α-β)e^-2sg(ζ)                    1 ],                                                                                        ζ∈Σ_1,;         [                   1                   0; e^π i(α-β)e^-2sg(ζ)                   1 ],                                                                                        ζ∈Σ_2,; [                     1 -e^-π i(α-β) e^2sg(ζ);                     0                     1 ],                                                                                        ζ∈Σ_3,;                                                                                  e^2π iβσ_3 ,                                                                                        ζ∈Σ_4,;     [                    1 -e^π i(α-β) e^2sg(ζ);                    0                    1 ],                                                                                        ζ∈Σ_5,;                                         [           0 -e^π i(α-β); e^-π i(α-β)           0 ],                                                                                        ζ∈Σ_6,;                                     [            0 -e^-π i(α-β);   e^π i(α-β)            0 ],                                                                                        ζ∈Σ_7. ] .





  * (c) As ζ→∞,  we have
  
    A(ζ)=( I +
    O(1/ζ) ) ζ^-βσ_3,

 where
 ζ∈ (-π/2, 3π/2).


 
  * (d) The behaviors of A(ζ) near ±1 and the origin are the same as Ψ(ζ;s); cf. (<ref>), (<ref>) and (<ref>).
  




 §.§ Outer parametrix


As is→+∞, the jump matrices (<ref>) on the contours Σ_k, k=1,2,3,5 tend to the identity matrix exponentially fast.
Thus, we consider the following approximate RH problem with jumps on the remaining contours.



  §.§.§ RH problem for A^(∞)


  
  * (a) A^(∞)(ζ)
 is analytic in
  ℂ∖{Σ_4∪Σ_6∪Σ_7}; see Figure <ref> for the contours.
  
  * (b)  A^(∞)(ζ) satisfies the jump conditions
  
    A^(∞)_+(ζ)=A^(∞)_-(ζ)
     {[                                              e^2π iβσ_3 ,                                                    ζ∈Σ_4,;     [           0 -e^π i(α-β); e^-π i(α-β)           0 ],                                                    ζ∈Σ_6,; [            0 -e^-π i(α-β);   e^π i(α-β)            0 ],                                                    ζ∈Σ_7. ] .





  * (c) As ζ→∞,  we have
  
    A^(∞)(ζ)=( I +
    O(1/ζ) ) ζ^-βσ_3,

 where
 ζ∈ (-π/2, 3π/2).
  

A solution can be constructed explicitly, that is,

    A^(∞)(ζ)=2^βσ_3e^-β/2π iσ_3( I-iσ_1 /√(2))( ζ-1 /ζ+1)^1/4σ_3( I+iσ_1 /√(2)) φ(ζ)^-βσ_3e^β/2π iσ_3D(ζ)^σ_3,

   where φ(ζ)=ζ+√(ζ^2-1) is defined as before  in (<ref>).
 The branches for √(ζ^2-1) and ( ζ-1 /ζ+1)^1/4 are taken such that  (ζ± 1)∈(-π, π) and the branch for ζ^β is  taken such that ζ∈(-π/2, 3π/2).
The function   D(ζ) is the Szegő function which solves  the following scalar RH problem:

  
  * (a) D(ζ) is analytic in ℂ∖ [-1,1].

  * (b) D(ζ) satisfies the relation

    D_+(x)D_-(x)= {[   e^π iα,   -1<x<0,; e^-π i α,    0<x<1. ].


Applying the Sokhotski-Plemelj formula, we have

    D(ζ)=exp{α√(ζ^2-1)/2i  (-∫_-1^0 1/√(1-x^2)dx/ζ-x
    +∫_0^11/√(1-x^2)dx/ζ-x)}=(-i+√(ζ^2-1)/ζ)^α,

where the   branches for √(ζ^2-1) and ζ^α are taken such that (ζ±1), ζ∈  (-π,π); see <cit.>.
Straightforward calculation from (<ref>) gives

    D(ζ)=1-iα/ζ+O(1/ζ^2),    ζ→∞,


    D(ζ)=e^-iαπ/2 (ζ/2)^α(1+O(ζ) ),   ζ→ 0,   ζ>0,

and

    D(ζ)=e^∓ iαπ/2(1+O(√(ζ^2-1))),   ζ→± 1,   ζ>0.

Substituting (<ref>) into (<ref>), we have

    A^(∞)(ζ)=2^βσ_3e^-β/2π iσ_3(  I-iσ_2/√(2) ) 2^-ασ_3 e^-1/2π iασ_3(I+O(ζ))ζ^ασ_3,   ζ→ 0,   ζ>0,

where the Pauli matrix σ_2 is defined in (<ref>). The behavior of A^(∞)(ζ) as ζ→ 0 from the lower-half plane can be obtained by combining (<ref>) with (<ref>).
Similarly, we get by substituting (<ref>) into (<ref>)

    A^(∞)(ζ)=2^βσ_3e^-β/2π iσ_3(I+-iασ_3+1/2σ_2/ζ+O(1/ζ^2)) 2^-βσ_3 e^1/2π iβσ_3ζ^-βσ_3,   ζ→∞,

for ζ∈(-π/2,3π/2).




 §.§ Local parametrices


We need the local parametrices A^(+1)(ζ) and  A^(-1)(ζ) near ζ= 1 and ζ=- 1, respectively.

  
  * (a) A^(±1)(ζ) is analytic respectively in U(±1)∖{∪^7_j=1Σ_j}, where Σ_j are illustrated in Figure <ref>, and U(±1) are small disks centered at ±1, respectively.

  * (b) A^(±1)(ζ) satisfy the same jump conditions as A(ζ) in U(±1)∩{∪^7_j=1Σ_j}.

  * (c) A^(±1)(ζ) satisfy the matching condition

    A^(±1)(ζ)=(I+O(1/s))A^(∞)(ζ)

for ζ on the boundaries of U(±1), as is→ +∞.


We focus on the construction of the local parametrix near ζ=-1. The  local parametrix near ζ=1 can be
constructed similarly. In   the neighborhood U(-1), we seek a local parametrix of the   form

    A^(-1)(ζ)=E_-1(ζ) Φ_B(-|s|^2g^2(ζ) ){[       e^1/2π i(α-β)σ_3e^-sg(ζ)σ_3,                               ζ>0,; iσ_2 e^1/2π i(α-β)σ_3 e^-sg(ζ)σ_3,                               ζ<0, ].

where E_-1(ζ) is defined and analytic in   U(-1). From the definition (<ref>) it is readily seen that
-g^2(ζ) serves as a conformal mapping in U(-1) and U(+1). For later use, we   assign

    √(-g^2(ζ))= {[ -ig(ζ),    ζ>0,;  ig(ζ),    ζ<0, ].

with the ζ-plane being cut   along (-∞, -1]∪ [1, +∞), and the square root   being real positive for -1<ζ<1; cf. (<ref>).
 The function  Φ_B(ζ) in  (<ref>) solves the following model RH problem.





  
  * (a)   Φ_B(ζ) is analytic in
  ℂ\Σ_B, where the contour  Σ_B is illustrated in Figure  <ref>.

  
  * (b)   Φ_B(ζ)  satisfies the jump condition
  
    Φ_B+(ζ)=Φ_B-(ζ) J_B(ζ),   ζ∈Σ_B,

where the jump  J_B(ζ) is also indicated in Figure  <ref>.
  
  * (c)  The asymptotic behavior of Φ_B(ζ)  at infinity is
  
    Φ_B(ζ)=ζ^-1/4σ_3I+iσ_1/√(2) (I+1
       /8√(ζ)[  -1 -2i; -2i   1 ]+O (1/ζ ))e^√(ζ)σ_3  as ζ→∞ .



The solution to the above RH problem can be constructed in terms of Bessel functions

    Φ_B(ζ)=π^1/2σ_3{[                                                                                (
                                   [             I_0(√(ζ))         i/π K_0(√(ζ)); π i√(ζ)    I'_0(√(ζ))   -√(ζ)    K'_0(√(ζ));                       ]),                                                                                                                                                                                                                                  ζ∈I,;                                                              (
                                   [             I_0(√(ζ))         i/π K_0(√(ζ)); π i√(ζ)    I'_0(√(ζ))   -√(ζ)    K'_0(√(ζ));                       ])  [  1  0; -1  1 ],                                                                                                                                                                                                                                 ζ∈II,; (
                                   [             I_0(√(ζ))         i/π K_0(√(ζ)); π i√(ζ)    I'_0(√(ζ))   -√(ζ)    K'_0(√(ζ));                       ])
                                 (
                                   [ 1 0; 1 1;   ]),                                                                                                                                                                                                                                ζ∈III, ] .

where ζ∈ (-π, π); see <cit.>.  For later use, we mention that Φ_B satisfies the differential equation

    dΦ_B(ζ)/dζ=(
                                   [     0 1/2iζ;   i/2     0;       ])Φ_B(ζ).



To satisfy the matching condition, for ζ∈ U(-1), we define  E_-1(ζ) as

    E_-1(ζ)=A^(∞)(ζ)
    {[        e^-1/2π i(α-β)σ_3I-iσ_1/√(2) (-|s|^2g^2(ζ) )^1/4σ_3,                                                        ζ>0,; e^-1/2π i(α-β)σ_3(-iσ_2)I-iσ_1/√(2) (-|s|^2g^2(ζ) )^1/4σ_3,                                                        ζ<0, ].

where g(ζ) and A^(∞)(ζ) are defined in (<ref>) and (<ref>), respectively.
From (<ref>) and (<ref>), we see that E_-1(ζ) is analytic for ζ∈ U(-1). Particularly, in view of (<ref>) and (<ref>), we have

    E_-1(-1)=2^βσ_3e^-1/2βπ iσ_3I-iσ_1/√(2)|s|^1/2σ_32^-1/2σ_3e^1/4π iσ_3.



The other parametrix A^(+1)(ζ) at ζ=1,   in a ζ-neighborhood  U(+1), can be constructed similarly, also in terms of the Bessel-type model RH problem Φ_B. More precisely, we define

    A^(+1)(ζ)=E_1(ζ)σ_3 Φ_B(-|s|^2g^2(ζ))σ_3{[       e^-1/2π i(α-β)σ_3e^-sg(ζ)σ_3,                                 ζ>0; iσ_2 e^-1/2π i(α-β)σ_3 e^-sg(ζ)σ_3,                                ζ<0, ].

where

    E_1(ζ)=A^(∞)(ζ)
    {[        e^1/2π i(α-β)σ_3I+iσ_1/√(2) (-|s|^2g^2(ζ))^1/4σ_3,                                                      ζ>0,; e^1/2π i(α-β)σ_3(-iσ_2)I+iσ_1/√(2) (-|s|^2g^2(ζ))^1/4σ_3,                                                      ζ<0, ].

and A^(∞)(ζ) is given in (<ref>).
Using (<ref>), (<ref>), (<ref>) and (<ref>), we see that E_1(ζ) is analytic for ζ∈ U(+1). Moreover, we have

    E_1(1)=2^βσ_3e^-1/2βπ iσ_3I+iσ_1/√(2)|s|^1/2σ_32^-1/2σ_3e^-1/4π iσ_3.





 §.§ Final transformation

Next, we define

    B̂(ζ)={[   A(ζ)A^(∞)(ζ)^-1,  ζ∖{U(+1)∪ U(-1)},; A(ζ)A^(± 1)(ζ)^-1,         ζ∈ U(± 1). ].

 For ζ∈∂ U(±1), clockwise-oriented, and ζ>0, we have jumps

    J_B̂(ζ)=A^(±1)(ζ)A^(∞)(ζ)^-1 .

As is→+∞, by a standard analysis of small-norm RH problems <cit.>, we have

    B̂(ζ)=I+B̂_1(ζ)/s+O(1/s^2 ),   is→ +∞,

where

    B̂_1(ζ)=1/2π i∮_∂ U(-1)J_B̂,1(x)dx/x-ζ+1/2π i∮_∂ U(+1)J_B̂,1(x)dx/x-ζ

for ζ∈ℂ∖{∂ U(-1)∪∂ U(+1)}, and the integration paths in (<ref>) are small clockwise circles encircling -1 and 1, respectively.
Here, J_B̂,1 is the coefficient taken from

    J_B̂(ζ)=I+J_B̂,1(ζ)/s+O(1/s^2 ),   ζ∈∂ U(-1)∪∂ U(+1),   is→+∞,

where, for ζ∈∂ U(-1),

    J_B̂,1(ζ)=1/2√(ζ^2-1)A^(∞)(ζ)e^-1/2π i(α-β)σ_3[  -1 -2i; -2i   1 ]
    e^1/2π i(α-β)σ_3A^(∞)(ζ)^-1,

as follows from  (<ref>), (<ref>) and (<ref>), and a combination of (<ref>), (<ref>) and (<ref>) gives

    J_B̂,1(ζ)=1/2√(ζ^2-1)A^(∞)(ζ)e^-1/2π i(α-β)σ_3(
                         [ -1 2i; 2i  1;    ])e^1/2π i(α-β)σ_3A^(∞)(ζ)^-1

for ζ∈∂ U(+1),  with the principal branches taken such that (ζ±1)∈(-π,π).
Putting in use (<ref>) and (<ref>), we obtain that J_B̂,1(ζ) has simple poles  ζ=± 1 with the following leading term expansions
 
    J_B̂,1(ζ)∼1/4(ζ+1)2^βσ_3e^-β/2π iσ_3(σ_3-iσ_1)2^-βσ_3e^β/2π iσ_3    ζ→-1,

and

    J_B̂,1(ζ)∼1/4(ζ-1)2^βσ_3e^-β/2π iσ_3(σ_3+iσ_1)2^-βσ_3e^β/2π iσ_3    ζ→ 1.

Now the residue theorem applies. From (<ref>)  and  (<ref>)  we have

    1/2π i∮_∂ U(-1)J_B̂,1(x)dx/x-ζ=1/4(σ_3-i2^βσ_3e^-β/2π iσ_3σ_12^-βσ_3e^β/2π iσ_3)1/ζ+1

for ζ∉U(-1), and

    1/2π i∮_∂ U(+1)J_B̂,1(x)dx/x-ζ=1/4(σ_3+i2^βσ_3e^-β/2π iσ_3σ_12^-βσ_3e^β/2π iσ_3)1/ζ-1

for ζ∉U(+1).
Then, substituting (<ref>) and (<ref>) into (<ref>) gives the approximation as ζ→∞

    B̂_1(ζ)= σ_3/2ζ+O(1/ζ^2).






 §.§ Proof of Theorem <ref>: Asymptotics of sH as is→ +∞


In the outer region |ζ-1|>δ and |ζ+1|>δ for some small positive constant δ, we get from the transformations (<ref>) and  (<ref>)
 
    Ψ(ζ)=B̂(ζ)A^(∞)(ζ)exp(s/4√(ζ^2-1) σ_3).

 Expanding the last term as ζ→∞, we obtain
 
    exp(s/4√(ζ^2-1) σ_3)=exp(1/4sζσ_3) (I-s/8ζσ_3+O(1/ζ^2)).

Substituting this and the expansions (<ref>) and (<ref>) into (<ref>), and recalling (<ref>), we
have
 
    H(s)=-(Ψ_1)_11/2-(α^2-β^2)/s=s/16+iα/2-
    (α^2-β^2+1/4)1/s+O(1/s^2)
 as is→+∞. This completes the proof of  (<ref>).


From  (<ref>), (<ref>), (<ref>), (<ref>)  and (<ref>), we obtain

    y(s)=2^2βe^-βπ i(1+ O(  1/s)),    is→+∞,

and

    d(s)=2^-2α e^-απ ie^is/2α (1+O(1/s)),    is→+∞.

Now we derive the large-s behavior of u_k and v_k introduced in the Hamiltonian (<ref>).
In the region |ζ+1|<δ, we get by tracing back the invertible transformations (<ref>) and (<ref>)
  
    Ψ(ζ)=B̂(ζ)E_-1(ζ) Φ_B (-|s|^2g^2(ζ) )
    e^1/2π i(α-β)σ_3,  ζ>0,

where E_-1(ζ) is independent of s and analytic for |ζ+1|<δ,
B̂(ζ) is also analytic for |ζ+1|<δ.
Thus, we have

    A_2(s)   =B̂(-1)E_-1(-1) (lim_ζ→ 0ζΦ_B'(ζ)Φ_B^-1(ζ))E_-1(-1)^-1B̂(-1)^-1
       =-i/2sB̂(-1)E_-1(-1) σ_+E_-1(-1)^-1B̂(-1)^-1
       =i/8sB̂(-1)2^βσ_3e^-1/2βπ iσ_3(iσ_3+σ_1)
     e^1/2βπ iσ_32^-βσ_3B̂(-1)^-1,

where use has been made of the differential equation (<ref>) and formula (<ref>).
It follows from (<ref>) that

    B̂(-1)=I+O(1/s),   is→+∞.

Thus, from (<ref>), (<ref>) and (<ref>)-(<ref>),  we obtain the asymptotic approximations of u_2(s) and v_2(s)  as is→+∞, as stated in
(<ref>) and (<ref>).

Using the approximation of Ψ(ζ;s) for |ζ-1|<δ, a similar argument justifies (<ref>) and (<ref>), namely,   the asymptotic approximations of u_1(s) and v_1(s)   as is→+∞.

Thus, we have obtained the  large-s asymptotics in Theorem <ref>.
This, together with the  small-s asymptotic formulas obtained in  Section <ref> and
the existence results given in Proposition <ref>,  completes the proof of Theorem <ref>.



It is worth noting that with   long but direct calculation   the later terms in the asymptotic expansions of u_k(s) and v_k(s) can also be determined.
For example, we have from (<ref>) that

    B̂(±1)=I+B̂_1(±1)/s+O(1/s^2),    is→+∞.

Thus, the asymptotics (<ref>) and (<ref>) can be refined
respectively to

    v_1(s)=i+γ_1/s+O(1/s^2),    is→+∞,

and

    v_2(s)=-i+γ_2/s+O(1/s^2),    is→+∞,

where the coefficients γ_1 and γ_2 can be determined by using B̂_1(±1).





§ RH PROBLEM FOR ORTHOGONAL POLYNOMIALS AND THE DIFFERENTIAL IDENTITY

We consider the orthogonal polynomials associated with the weight function (<ref>).
Note that the weight function (<ref>) can be analytically  extended to the complex z-plane with a cut  [0,∞)
 
    w(z)=(z-1)^2αz^β-αe^-π i(α+β),

where the branch cut of (z-1)^2α is taken along [1,∞)
 so that (z-1)∈(0,2π) and the branch of z^β-α is chosen so that z∈ (0,2π).
 Let π_n(z;t)=z^n+⋯
  be the monic orthogonal polynomial  of degree n with respect to
  the weight (<ref>), satisfying the orthogonality relation
 
    1/2π∫_t^2π-tπ_n(e^iθ;t)π_m(e^iθ;t)w(e^iθ;t)dθ=χ_n^-2(t)δ_n,m,

where χ_n(t)>0 and n,m=0,1,⋯. Denote the reverse polynomials  associated with π_n(z) by

    π_n^*(z)=z^nπ_n(1/z̅)=z^nπ_n(1/z)=z^n+⋯,
where π_n denotes the polynomial whose coefficients are complex conjugates of that of π_n.
Let Y(z) be the  2× 2 matrix-valued function

    Y(z)=
    [                                       π_n(z)           1/2π i ∫_C_tπ_n(x) w(x)dx/x^n(x-z);                          -χ_n-1^2 π^*_n-1(z) -χ_n-1^2/2π i∫_C_tπ^*_n-1(x) w(x)dx/x^n(x-z) ].

Then,  Y(z) is the unique solution to the RH problem below.


  §.§.§ RH problem for Y


  
  * (a)  Y(z) is analytic in ℂ∖ C_t.

  
  * (b) Y(z)  satisfies the jump condition
  
    Y_+(z)=Y_-(z)
      [           1 z^-nw(z;t );           0           1 ],     z∈ C_t,

where  the arc C_t is defined in (<ref>). The weight  function w(z;t), as given  in (<ref>),  is the restriction of w(z) (<ref>) on the arc C_t.
  
  * (c)  As z→∞, we have
  
    Y(z)= (I+O (1/z ))
     [  z^n    0;    0 z^-n ].

 
  * (d)   Denoting  z_1=e^it and z_2=e^i(2π-t), we have, respectively  for   z_*=z_1 and  z_*=z_2, that
  
    Y(z)=[         O(1) O(ln(z-z_*));         O(1) O(ln(z-z_*)) ]    z→ z_*.



We have the  differential identities with respect to t in the following lemma, involving Y(z) and the corresponding Toeplitz determinant. The lemma was first proved in <cit.>.

Let D_n(t) be the Toeplitz determinant associated with the weight function (<ref>), then we have

    d/dtln D_n(t)=-1/2π∑_j=1^2z^-n+1_j w(z_j;t)lim_z→ z_j(Y^-1(z)d/dz Y(z))_21,

where z_1= e^it and z_2= e^i(2π-t).  

We start with the  expression of the Toeplitz determinant  in terms of the leading coefficients of the orthonormal polynomials

    D_n(t)=∏_k=0^n-1χ_k^-2(t).

Taking logarithmic derivative on both side of (<ref>) yields

    d/dtln D_n(t)=-2∑_k=0^n-1χ_k^-1(t)d/dtχ_k(t).

The orthogonal relation (<ref>)
serves as an integral representation  of χ_n(t). Taking derivatives on both sides, we have
 
    2χ_k^-1(t)d/dtχ_k(t)=χ_k^2(t)/2π∑_j=1^2 π_k(z_j)π_k(z^-1_j)w(z_j;t),

where z_1= e^it, z_2= e^i(2π-t) and k=0,1,...,n-1.
Substituting (<ref>) into (<ref>) gives

    d/dtln D_n(t)=-1/2π∑_j=1^2(∑_k=0^n-1χ_k^2(t)π_k(z_j)π_k(z^-1_j)w(z_j;t)).

From
the Christoffel-Darboux identity <cit.>

    ∑_k=0^n-1χ_k^2(t)π_k(z)π_k(z^-1)=zχ_n^2(t)[π_n(z^-1)d/dzπ_n(z)-π_n(z) d/dzπ_n(z^-1)]
    -nχ_n^2(t)π_n(z)π_n(z^-1),

and the recurrence relation <cit.>

    zπ_n(1/z)=χ_n-1(t)^2/χ_n(t)^2π_n-1(1/z)+
    z^1-nπ_n(0)π_n(z),

we have

    ∑_k=0^n-1χ_k^2(t)π_k(z)π_k(z^-1)
       =-χ_n-1^2(t)[π_n(z) d/dzπ_n-1(z^-1)
    -π_n-1(z^-1)d/dzπ_n(z)+(n-1)z^-1π_n(z)π_n-1(z^-1)]
       =-χ_n-1^2(t)z^1-n[π_n(z) d/dzπ_n-1^*(z^-1)
    -π_n-1^*(z^-1)d/dzπ_n(z)].

Then, the differential identity  (<ref>) follows from (<ref>), (<ref>) and (<ref>).







§ ASYMPTOTICS OF THE RH PROBLEM FOR Y

We carry out the Deift-Zhou nonlinear steepest descent  analysis of the RH problem for Y(z) as n→∞ and t→ 0^+ in a way such that nt is bounded.


 §.§ Normalization of the RH problem: Y→ T

We introduce the transformation Y→ T to normalize the large-z behavior of Y(z),

    T(z)={[ Y(z)z^-nσ_3,       |z|>1,;        Y(z),       |z|<1, ].

Then, it is readily seen that T(z) satisfies the following RH problem.
 

  §.§.§ RH problem for T


  
  * (a)  T(z) is analytic in ℂ∖ C_t.

  
  * (b) T(z)  satisfies the jump condition
  
    T_+(z)=T_-(z)
      [  z^n w(z);    0 z^-n ],     z∈ C_t,

  and
  
    T_+(z)=T_-(z)
      [  z^n    0;    0 z^-n ],     z∈ C∖ C_t,

  where  the weight function w(z) is defined in (<ref>), C={z: |z|=1} is the unit circle and the arc C_t is defined in (<ref>). Both the arcs in (<ref>) and (<ref>) are oriented counterclockwise.
       
  * (c)  As z→∞, we have
  
    T(z)=I+O (1/z ).

 
  * (d)   As z→ z_1 and z→ z_2, where z_1=e^it and z_2=e^i(2π-t), T(z) shares the same behavior as Y(z); cf. (<ref>).




 §.§ Opening of the lens: T→ S

The diagonal entries of the jump matrix for T  in (<ref>) are highly oscillating for n large.
 To turn the oscillation to exponential decays on certain contours, we deform the contours C_t and introduce
 the second transformation T→ S.
The transformation is  based on the following factorization of the  jump matrix  (<ref>)
 
    [   z^n w(z );     0  z^-n ]=  [           1           0; z^-nw(z)^-1           1 ][         0      w(z); -w(z )^-1         0 ][          1          0; z^nw(z)^-1          1 ].

  Define the transformation
  
    S(z)={[ T(z) [            1            0; z^-nw(z )^-1            1 ] ,                                                      for z∈Ω_E,;     T(z)[           1           0; -z^nw(z)^-1           1 ]  ,                                                      for z∈Ω_I,;                                                           T(z),                                                      otherwise,;                                                                 ].

 with the regions shown in Figure <ref>.
 Then,  S(z) satisfies the following RH problem.
  






 

  §.§.§ RH problem for S


  
  * (a)  S(z) is analytic in ℂ∖Σ_E∪ C∪Σ_I, where the jump curves are illustrated in Figure  <ref>, and C= {z:|z|=1} is the unit circle.

  
  * (b) S(z)  satisfies the jump condition
  
    S_+(z)=S_-(z){[ [           1           0; z^-nw(z)^-1           1 ]  ,                                              for z∈Σ_E,;              [        0     w(z); -w(z)^-1        0 ] ,                                             for z∈ C_t,;     [          1          0; z^nw(z)^-1          1 ]  ,                                              for z∈Σ_I,;                              [  z^n    0;    0 z^-n ] ,                                          for z∈ C∖ C_t. ].

  
  * (c)  As z→∞, we have
  
    S(z)=I+O (1/z ).

 
  * (d)   At z_1=e^it and z_2=e^i(2π-t),   we have
  
    S(z)=O(ln(z-z_1))     z→ z_1       S(z)=O(ln(z-z_2) )     z→ z_2 .





 §.§ Global parametrix

As n→∞, the jump matrices in (<ref>) tend  to the identity matrix for z bounded away from the unit circle  C.
The  arc C_t also tends to the unit circle in the double scaling case as n→∞.   Thus, we consider the approximate RH problem below, with jump occurs on the unit circle alone.



  §.§.§ RH problem for N


  
  * (a)  N(z) is analytic in ℂ∖   C, where C is the unit circle.

  
  * (b) N(z)  satisfies the jump condition on the unit circle, counterclockwise oriented,
  
    N_+(z)=N_-(z) [        0     w(z); -w(z)^-1        0 ],
where the weight w(z) is given in (<ref>).
  
  * (c)  As z→∞, we have
  
    N(z)=I+O (1/z ).

 

 The solution to the above RH problem is given explicitly by
 
    N(z)={[                             (z-1/z)^(β-α)σ_3,                                        |z|>1,; (z-1)^(α+β)σ_3e^-π i(α+β)σ_3[  0  1; -1  0 ],                                        |z|<1, ].

 where the branch of z^α-β is chosen so that z∈ (0,2π) and both branch cuts of (z-1)^α+β
 and (z-1)^β-α  are taken along [1,∞), with (z-1)∈(0,2π).

 

 §.§ Local parametrix

The global parametrix N(z) fails to   approximate   S(z) near the endpoints z_1 and z_2, where the weight has jump discontinuities.  For n large,
 both the endpoints   are close to 1 and  belong to a neighborhood U(1,r)={z: |z-1|<r} of z=1, with r small but fixed.  We intend to
 construct a local parametrix P^(1)(z) in   U(1,r), which  satisfies the same jump condition as S(z) and
  matches with N(z) on the boundary of  U(1,r).


   

  §.§.§ RH problem for P^(1)


  
  * (a)  P^(1)(z) is analytic in U(1,r) ∖{Σ_E∪ C∪Σ_I}; see Figure <ref> for the contours.

  
  * (b) P^(1)(z)  satisfies the same jump condition as S(z) on  U(1,r) ∩{Σ_E∪ C∪Σ_I}; cf. (<ref>).
   
  * (c)  As n→∞, we have the matching condition
  
    P^(1)(z)=(I+O (1/n ))N(z).



Let

    M(z)={[                z^n/2σ_3w(z)^-1/2σ_3,                    |z|<1, z∈ U(1,r),; z^n/2σ_3w(z)^1/2σ_3[  0 -1;  1  0 ],                    |z|>1, z∈ U(1,r), ].
where the principal branches are chosen for the power functions.
  We also define
  
    f(z)=f(z;t)=e^-π i/2 t^-1ln z,

where the branch   of  ln z  is chosen such that z∈ (-π, π).
Note that f(z) is a conformal mapping from  U(1,r) to a neighborhood of the origin so long as 0<r<1, with f(z_1)= 1 and  f(z_2)= -1.
  We seek a solution to the above RH problem of the following form:

    P^(1)(z)=E(z)Ψ(f(z); -2int )e^±π i/2(α-β)σ_3M(z),    ± z>0,

 where Ψ(ζ; s) is the solution to the model RH problem discussed in Section <ref>,  E(z) is a certain matrix-valued analytic function defined in U(1,r), to be determined.
It then follows from the jumps (<ref>) for Ψ and the definition of M(z) in (<ref>)  that P^(1)(z) fulfills the same jump condition  as S(z) for z ∈ U(1,r)∩{Σ_E∪ C∪Σ_I}.
We then choose E(z) so that  the matching condition  (<ref>) holds.


Let

    E(z)=e^±π i(α-β)/2σ_3 e^-π i(α+β)/2σ_3 z^α-β/2σ_3(f(z)/z-1)^-βσ_3[  0  1; -1  0 ], ± z>0,  z∈ U(1,r),

where r∈ (0, 1) is a small constant, the function ζ^-β takes  the principal branch, and the branch of z^(α-β)/2 is chosen such that z∈ (0, 2π).
Then, E(z) is analytic for z∈ U(1,r) and  the matching condition  (<ref>) is satisfied for nt bounded.

 It follows from the definition of f(z) in (<ref>) that
(f(z)/z-1)^-β=e^π i β/2 t^β+O(z-1) is analytic for z∈ U(1,r).
With the branch chosen, it is readily seen that e^±π i(α-β)/ 2     z^ (α-β)/2
 is also analytic  for z ∈ U(1,r) with r<1. Thus, E(z) is  analytic     in U(1,r).

Next, we show that P^(1)(z) satisfies the  matching condition (<ref>). Since nt is bounded, we see that f(z) is large for n large. Hence, for   z∈∂ U(1,r), in view of (<ref>) and   the large-ζ behavior (<ref>) of Ψ(ζ; s), we obtain

    P^(1)(z)N(z)^-1=E(z)[I+O(  1 /n) ]f(z)^-βσ_3z^-n/2σ_3
    e^±π i(α-β)/2σ_3M(z)N(z)^-1,   ± z>0.

Recalling the definitions of N(z) and M(z), given  in (<ref>) and (<ref>), respectively, we have

    M(z)N(z)^-1=z^n/2σ_3e^-π i/2(α+β)σ_3z^1/2(α-β)σ_3(z-1)^βσ_3[  0 -1;  1  0 ],

where the branch for  z^n/2  is chosen such that z∈ (-π, π), the branch cut for z^ (β-α)/2 is taken along the positive real axis so that z∈ (0,2π) and
the branch cut for (z-1)^-β is chosen along [1,∞ ) so that (z-1)∈ (0,2π). Note that |f(z)^β|=e^-π i/2βe^iβln z is independent  of t, bounded and
bounded away from zero for pure imaginary parameter β and  z ∈∂ U(1,r).
Substituting (<ref>) and (<ref>)  into (<ref>) gives us the matching condition (<ref>).
This completes the proof of the proposition.





 §.§ Final transformation: S → R


Define

    R(z)={[     S(z)N(z)^-1,         |z-1|>r,; S(z)P^(1)(z)^-1,         |z-1|<r. ].

 Then, R(z) satisfies the RH problem.





  

  §.§.§ RH problem for R


  
  * (a)  R(z) is analytic for  C∖Σ_R, where the remaining contour  Σ_R is illustrated in Figure <ref>.

  * (b) R(z)  satisfies the   jump condition
  
    R_+(z)=R_-(z)J_R(z),    z∈Σ_R,

with

    J_R(z)={[    P^(1)(z)N(z)^-1, |z-1|=r, clockwise,;  N(z)J_E(z)N(z)^-1,     z∈Σ_E, |z-1|>r,;  N(z)J_I(z)N(z)^-1,     z∈Σ_I, |z-1|>r, ].
   where J_E(z) and J_I(z) denote the jumps of S(z) on Σ_E and Σ_I, respectively; see (<ref>) for the jump matrices and Figure <ref> for the contours.

  * (c)  As z→∞, we have
  
    R(z)=I+O(1/z).



It is readily seen that J_R(z)-I is uniformly exponentially small for z on the portions of
 Σ_E and Σ_I, described in (<ref>), as n→∞.
 Therefore, for large n and bounded nt, we have from the matching condition (<ref>) that
 
    J_R(z)=I+O(1/n),

 uniformly for z∈Σ_R.
As before, after a standard analysis of the small-norm RH problems <cit.>, a combination of  (<ref>) and (<ref>) gives
 
    R(z)=I+O(1/n),    d/dzR(z)=O(1/n),

uniformly in the whole complex   z-plane.


 §.§ Proof of Theorem <ref>


As a consequence of the nonlinear steepest descent analysis of the RH problem for Y performed in Section <ref>,  and applying the differential identity (<ref>), we derive  the asymptotic approximation of the Toeplitz determinant as stated in Theorem <ref>.

Tracing back the series of transformations Y→ T→ S→ R, we find that
 
    Y(z)=R(z) E(z)Ψ (f(z); -2int )e^±π i(α-β)/2σ_3z^n/2σ_3w(z)^-1/2σ_3,  ± z>0,

for  |z|<1 and |z-1|<r.
From Proposition <ref>, we see that E(z) is analytic and bounded for |z-1|<r.  We also have the analyticity of
R(z) in |z-1|<r and  the approximation (<ref>).
Thus, we have
 
    z^-nw(z)(Y(z)^-1d/dz Y(z))_21=f'(z)e^±π i(α-β){Ψ^-1(f(z); -2int )Ψ_ζ(f(z); -2int )}_21+O(1/n),
respectively for ± z>0,
with  |z|<1  and |z-1|<r.
The definition of f(z) implies that f'(z)=1/itz; cf. (<ref>).
Substituting (<ref>) into the differential identity (<ref>), we obtain

    d/dtln D_n(t)=   -1/2π i te^π i(α-β)lim_ζ→ 1(Ψ_+(ζ; -2int )^-1d/dζΨ_+(ζ; -2int ))_21
       -1/2π i t e^ -π i(α-β)lim_ζ→ -1(Ψ_+(ζ; -2int )^-1d/dζΨ_+(ζ; -2int ))_21+O(1/n )
    
    =    -1/2π i t(e^π i(α-β)Ψ^(1)_1(-2int)+e^ -π i(α-β)Ψ_1^(2)(-2int))_21+O(1/n ),

where Ψ^(1)_1(s) and Ψ^(2)_1(s) are defined in (<ref>) and  (<ref>), respectively.
Comparing (<ref>) with (<ref>), we have

    -1/π i (e^π i(α-β)Ψ^(1)_1(s)+e^ -π i(α-β)Ψ_1^(2)(s))_21=2sH(s)+c,

where H(s) is the Hamiltonian for the coupled Painlevé V system and  c is a constant independent of the variable s. Using the asymptotics as is→0^+ in  (<ref>) and  (<ref>), we have the constant c=0.
Substituting (<ref>) into  (<ref>), we then obtain by taking integration

    D_n(t)/D_n(0)=exp(∫_0^-2intH(τ)dτ+O(1/n)),

where  the error bound is uniform for nt bounded. Thus we obtain (<ref>), and complete the proof of Theorem <ref>.





§ PROOF OF THEOREM <REF>   


Integrating  the large-s asymptotic approximation of H(s;α,β) in Theorem <ref>, we have as is→ +∞

    ∫_0^sH(τ;α,β)dτ=s^2/32+iα/2s-(α^2-β^2+1/4)ln|s|+C_1(α,β)+O(1/s),

where the path of integration is  a line segment   of the negative imaginary axis.
Since H(s) is free of poles for s∈ -i(0, +∞) with at most weak singularity as is→ 0^+, as indicated in the asymptotic behavior   (<ref>),  the integral  (<ref>)  is well-defined.

To determine the constant C_1(α,β),  which contains the global information of the Hamiltonian, the strategy is the following.
First, we express the integral alternatively

by using the differential identity (<ref>).
The evaluation of the constant C_1(α,β) amounts to a calculation of   the constant term C_2(α,β) in the asymptotic expansion of the action integral. Next, it is much more convenient  to evaluate    C_2(α,β).
Actually,  by  using the differential identities  (<ref>) and (<ref>), we see that each derivative  of the action integral, with respect to the parameters α or β, is a simple combination of the  functions u_k(s) and v_k(s), not involving    any integrals.  Thus, the derivatives of the constant term C_2(α,β) with respect to the parameters α and β  are then readily determined.  Finally, it is known that for special parameters α=1/2 and β=0, the Hamiltonian admits a special function solution (<ref>) and its integral is explicitly calculated in (<ref>).
This special case actually serves as an initial condition. The constant terms are thus determined, in terms of the Barnes G-function.


To integrate on both sides of the differential   identity (<ref>), we first consider the following limit
 as is→ 0^+,

    lim_is→ 0^+(sH(s;α,β)+αln d(s)-βln y(s)-2(α^2-β^2)ln s   )  
       = αln(Γ(1+α+β)Γ(1+α-β)/Γ(1+2α)^2)
    -βln(Γ(1+α-β)/Γ(1+α+β))   
         +αln(2α)-2(α^2-β^2)ln2,

by using (<ref>) and (<ref>),
and  the asymptotics of  H(s;α,β) given in Theorem <ref>.
 Recalling (<ref>), the convergence of the integral  (<ref>) and the    boundedness near 0^+ of the function in  (<ref>) imply that  the action differential defined by
 
    u_1(s)dv_1(s)/ds+u_2(s)dv_2(s)/ds-H(s;α,β)

 is pole-free and integrable along -i(0,δ_1]  for some small δ_1>0.
 From the large-s asymptotics of H(s;α,β), u_k(s), v_k(s) in Theorem <ref>, the action differential is also free of poles on -i[M_1,+∞) for   M_1>0 sufficiently large. Moreover, the action differential  has at most finite number of poles on -i(δ_1,M_1). Integrating  both sides of   (<ref>) on the interval [0,s], and using the approximation     (<ref>) as an initial value, we obtain
 
    ∫_0^sH(τ;α,β)dτ=   ∫_0^s[u_1(τ)dv_1(τ)/dτ+u_2(τ)dv_2(τ)/dτ-H(τ;α,β) ]dτ
       +sH(s;α,β)+αln d(s)-βln y(s)-2(α^2-β^2)ln s-C_0,

where  C_0 is the limit in (<ref>), the path of integration of the first integral is an interval of the negative imaginary axis as described  in (<ref>). The integration contour of the second integral consists of the intervals (0,-iδ_1], [-iM_1,s] and a curve Γ connecting δ_1 and M_1 in the right half complex plane τ>0 to avoid the poles of the action differential and the Hamiltonian, and such that the region inside the closed curve -i(δ_1,M_1)∪Γ^-1 contains no poles of the action differential  and the Hamiltonian.
 From the large-s asymptotic formulas (<ref>), (<ref>), (<ref>), (<ref>) and  (<ref>),  we obtain

    ∫_0^s[u_1(τ)dv_1(τ)/dτ+u_2(τ)dv_2(τ)/dτ-H(τ;α,β) ]dτ
       =-s^2/32-iα s/2
    +(α^2-β^2+1/4-i(γ_1+γ_2)/8)ln|s|+C_2(α,β)+
    O(1/s)

and

    sH(s;α,β)+αln d(s)-βln y(s)-2(α^2-β^2)ln s
       = s^2/16+i α s
    -2(α^2-β^2)ln |s|-2(α^2+β^2)ln 2
    -(α^2-β^2+1/4)+αlnα+O(1/s)

as is→+∞,  where the logarithms take the principal branches, the constant  C_2(α,β)  depends  only on the parameters α and β, and the constants γ_1 and  γ_2 are the coefficients in the asymptotic expansions (<ref>) and (<ref>), respectively.
In deriving  (<ref>), use has also been made of the large-s asymptotic formulas (<ref>) and (<ref>).
Substituting (<ref>), (<ref>) and  (<ref>) into (<ref>),  we obtain

    C_1(α,β)=   C_2(α,β)-(α^2-β^2+1/4 )-αln 2-4β^2ln2
       -αln(Γ(1+α+β)Γ(1+α-β)/Γ(1+2α)^2)+βln(Γ(1+α-β)/Γ(1+α+β)),

and the relation

    γ_1+γ_2=-4i.



Next, we evaluate the constant C_2(α,β) by using the differential identities (<ref>) and (<ref>).
From the relation (<ref>)  and
the large-s asymptotics of d(s), u_k(s) and v_k(s), given in (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we have

    u_1(s) dv_1(s)/dα +u_2 (s)dv_2(s)/dα -ln d(s) +2αln s=-i/2s+2αln|s|
    +2αln2
    -lnα+O(1/s),

and

    u_1(s) dv_1(s)/dβ +u_2(s) dv_2(s)/dβ +ln y(s) -2βln s=-2βln|s|+2βln2+O(1/s),

as is→ +∞.
On the other hand, from the asymptotics of  d(s), y(s), u_k(s) and v_k(s) as is→0^+, given in (<ref>), (<ref>) and Theorem <ref>, we have

    lim_is→0^+(u_1(s)dv_1(s)/dα+u_2(s)dv_2(s)/dα-ln d(s)+2αln s )
         =2αln 2-ln2αΓ(1+α+β)Γ(1+α-β)/Γ(1+2α)^2,

and

    lim_is→0^+(u_1(s)dv_1(s)/dβ+u_2(s)dv_2(s)/dβ+ln y(s)-2βln s)=
    -2βln 2+lnΓ(1+α-β)/Γ(1+α+β)
    .

Integrating   both sides of   (<ref>) and (<ref>) on the interval [0,s] (the integration contour deformed if necessary, similar to  (<ref>), to keep away from   the poles of the integrand), and using the approximations (<ref>) and (<ref>)-(<ref>), we obtain
the first-order differential equations

    d/dαC_2(α,β)=ln2+ln(Γ(1+α+β)Γ(1+α-β)/Γ(1+2α)^2),

and

    d/dβC_2(α,β)=4βln2-ln(Γ(1+α-β)/Γ(1+α+β)).

It follows from  (<ref>)  and  (<ref>) that

    C_2(α,β)=C_2(0,β)+αln2+∫_β^α+βln(Γ(1+τ))dτ+∫_-β^α-βln(Γ(1+τ))dτ-∫_0^2αln(Γ(1+τ))dτ,


    C_2(0,β)=C_2(0,0)+2β^2ln2+∫_0^βln(Γ(1+τ))dτ+∫_0^-βln(Γ(1+τ))dτ.

Recalling the integral expression of   the Barnes G-function

    ln G(1+z)=z/2ln (2π)-z(z+1)/2+zlnΓ(z+1)-∫_0^z lnΓ(1+t)dt,    z>-1;
cf., e.g., <cit.>,
we obtain from (<ref>) and (<ref>) that

    C_1(α,β)=   C_2(0,β)-ln(G(1+α+β)G(1+α-β)/G(1+2α))
       -4β^2ln2-1/4-∫_0^βln(Γ(1+τ))dτ-∫_0^-βln(Γ(1+τ))dτ.

This equation, together with (<ref>), implies that

    C_1(α,β)=C_2(0,0)-ln(G(1+α+β)G(1+α-β)/G(1+2α))
    -2β^2ln2-1/4.

In particular, we have

    C_1(1/2,0)=C_2(0,0)-ln(G(3/2)^2/G(2))-1/4=C_2(0,0)-ln(π G(1/2)^2)-1/4

by using the recurrence relation

    G(z+1)=Γ(z)G(z),    G(1)=1.

Comparing  (<ref>) and (<ref>), we obtain

    C_1(α,β)=C_1(1/2,0)+ln(π G( 1/2)^2G(1+2α)/G(1+α+β)G(1+α-β))-2β^2ln2.

Substituting (<ref>) into (<ref>), we obtain the large-s asymptotic approximation of the integral of  the Hamiltonian H(s)

    ∫_0^sH(τ;α,β)dτ=   s^2/32+iα s/2-(α^2-β^2+1/4)ln|s|+C_1(1/2,0)
       +ln(π G( 1/2)^2G(1+2α)/G(1+α+β)G(1+α-β))-2β^2ln2+O(1/s),

as is→+∞.


Finally,   we evaluate C_1(1/2,0) by using the special function solution to the Hamiltonian for α=1/2 and β=0.
From (<ref>), we have

    ∫_0^sH(τ;1/2,0)dτ=s^2/32+  is/4-ln|s|/2
    -ln(π/2)/2+O(1/s),    is→ +∞.
Here we have used the asymptotic formula I_0(x)=e^x/√(2π x){1+O(1/x)} as x→+∞; see <cit.>.
Comparing (<ref>) and (<ref>) yields

    C_1( 1/2,0)=1/2ln 2-1/2lnπ.

Thus, we obtain

    ∫_0^sH(τ;α,β)dτ=   s^2/32+iα s/2-(α^2-β^2+1/4)ln(|s|/4)
       +ln(√(π)  G( 1/2)^2 G(1+2α)/2^2α^2G(1+α+β) G(1+α-β))+O(1/s),

as is→+∞. The equation  can be written equivalently in the form of (<ref>). This completes the proof of Theorem <ref>.





§ ACKNOWLEDGEMENTS





The work of Shuai-Xia Xu was supported in part by the National Natural Science Foundation of China under grant numbers 11571376 and 11971492. Yu-Qiu Zhao was supported in part by the National Natural Science Foundation of China under grant numbers 11571375 and 11971489.



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