Starting from a generic initial solution u 0 (x) corresponding to parameters α, β, denoted as the triple (u 0 (x), α, β), we apply an explicit Bäcklund transformation to generate a family of solutions (u n (x), α + 4n, β + 4n) indexed by n ∈ N. We study the large n behavior of the solutions (u n (x), α + 4n, β + 4n) under the scaling x = z/n in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann-Hilbert representation of the solution u n (z/n). Our main result is a proof that the limit of solutions u n (z/n) exists and is given by a solution of the degenerate Painlevé-III equation, known as Painlevé-III(D 8 ),

A notable application of our result is to rational solutions of Painlevé-III(D 6 ), which are constructed using the seed solution (1, 4m, -4m) where m ∈ C \ Z + 1 2 and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at z = 0 when it is well defined, and by its monodromy data in the general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlevé-III, both D 6 and D 8 at z = 0. We also deduce the large n behavior of the Umemura polynomials in a neighborhood of z = 0.

This paper is a study of the confluence of solutions of the generic Painlevé-III equation to solutions of its parameter-free degeneration. The six Painlevé equations and their solutions, often referred to as Painlevé transcendents, have been the subject of intense study. This is largely motivated by the fact that Painlevé transcendents are generically transcendental, and yet appear in various applications in integrable systems, integrable probability, and random matrix theory to name a few.

All Painlevé equations but the first are actually families of differential equations indexed by complex parameters appearing as coefficients. However, certain solutions corresponding to different parameters can be related via Bäcklund transformations. For example, consider our main object of study, the generic Painlevé III equation, known as PIII(D 6 ):

In [17], Gromak discovered that the transformation u(x) → û(x) := 2xu ′ (x) + 4xu(x) 2 + 4xβu(x) -2u(x) u(x)(2xu ′ (x) + 4xu(x) 2 + 4x + αu(x) + 2u(x))

(1.2) mapped solutions of (1.1) with parameters (α, β) to solutions of (1.1) with parameters (α + 4, β + 4). With this one can construct from a given seed solution (u 0 , α, β) a family of solutions (u n , α+4n, β +4n) by iterating transformation (1.2). The paper [32] contains a survey of families of solutions of (1.1) constructed using this and other Bäcklund transformations. A notable family of solutions constructed in this manner is a sequence of rational solutions u = u n (x; m) obtained from the seed function u 0 (x) ≡ 1 and parameters α = -β = 4m. This family of solutions has been numerically and analytically explored in [5], and many conjectures were formulated there. While some of these were later resolved in the sequel [4], some conjectures remained open, see [5,Conjectures 4 and 5]. Conjecture 5 is concerned with the behavior of u n (x; m) near the singular point x = 0. As was done in [5], writing

and considering large n for fixed m yields the differential equation

Formally taking the limit and denoting the limiting function U (z; m) yields the parameter-free Painlevé-III equation, referred to as PIII(D 8 ),

The content of Conjecture 5 is that this convergence holds at the level of solutions, not just equations. , indicated in the plot with red/yellow circles. Likewise, all zeros of u 10 (x; 0.25) are simple with derivative 2/-2, indicated in the plot with pink/green squares. On the other hand, all poles and zeros of U (z; 0.25) have multiplicity 2 and are marked with red circles and green squares respectively.

To begin with, we prove Conjecture 5 from [5] in this work; to be more precise we establish the following theorem. for z / ∈ Σ(m), where Σ(m) denotes the union of all poles and zeros of z → U (z; m). The convergence is uniform on compact subsets of C \ Σ(m).

We illustrate this theorem in Figure 1. The pictures are made using the code from [10], which was generously provided by the authors.

In Section 2, we study the Maclaurin series solutions of (1.1); this characterizes the limiting solution of (1.3) via its initial conditions and produces a local version of Theorem 1.1, see Theorem 2.1 and Corollary 2.4 below.

The rational solutions u n (x; m) are related to the so-called Umemura polynomials s n (x; m) by the formula u n (x; m) = s n (x; m -1)s n-1 (x; m) s n (x; m)s n-1 (x; m -1) .

(1.5) Indeed, a sequence of rational functions x → s n (x; m) is determined by the recurrence relation It was shown in [8,42] that the rational functions s n (x; m) are actually polynomials. In Section 3, we use Corollary 2.4 to deduce asymptotics of the Umemura polynomials themselves. To formulate our result, we need to introduce a certain Fredholm determinant; more precisely, let K r : L 2 [0, r] → L 2 [0, r] denote the integral operator with the continuous Bessel kernel

For any λ ∈ C, let D λ (r) be the Fredholm determinant D λ (r) := det(1 -λK r ).

It is well known (see, e.g., [30,Chapter 24]) that the Fredholm determinant D λ (r) is an entire function of λ. Since K r is a trace-class integral operator, one of several equivalent ways to define D λ (r) is via the Plemelj-Smithies formula

Tr K ℓ r λ ℓ ℓ .

(1.8)

The traces in (1.8) have explicit expressions as iterated integrals

where K (1) (x, y) = K(x, y) and K (ℓ) (x, y) = r 0 K(x, t)K (ℓ-1) (t, y)dt.

By re-scaling the integrals to bring the r-dependence to the integrand and observing that J 0 √ xy and √ xyJ 1 √ xy are both entire functions with respect to both x and y, we see that Tr K ℓ r and D λ (r) can be extended to analytic functions of r in a neighborhood of r = 0, and in fact Tr K ℓ r = O r ℓ as r → 0, from which we obtain D λ (0) = 1. We are now ready to state our second theorem.

Theorem 1.2. Fix m ∈ C \ Z + 1 2 . Then, there exists a small enough neighborhood of the origin, G, such that the Umemura polynomials admit the following limits along the even and odd subsequences: D λ(m) (32iz), (1.10) where λ(m) = 1/ 1 + e 2πim , the square root and fractional powers denote the principal branches taking the value 1 at z = 0, and the convergence is uniform for z ∈ G. Furthermore, the values of the Umemura polynomials at the origin have the leading asymptotics s 2j (0; m) ∼ √ 2πe 4ζ ′ (-1) j 2j 2 +j+ m 2 2 + m 2 + 1 24 e -3j 2 -j 2 2j 2 +2j (-cos(πm)) j G 5 4

s 2j-1 (0; m) ∼ e 4ζ ′ (-1) √ 2π

24 e -3j 2 +j 2 2j 2 (cos(πm)) j G 3 4

in which G denotes the Barnes G-function and ζ denotes the Riemann zeta function.

In fact, one can check that the expressions on the right-hand side of (1.9) and (1.10) admit analytic continuation from a neighborhood of z = 0 to the whole z-plane. Although it does not follow from our proof, this suggests that the neighborhood G can be taken to be an arbitrary bounded set.

Our analysis of series solutions in Section 2 points to a more general statement about the coalescence of solutions of (1.1) to solutions of (1.3). The technical result leading to Theorem 1.1 by Maclaurin series (see Theorem 2.1 below) applies not only to rational solutions, but to all sequences of solutions with initial conditions converging to finite, nonvanishing limits. This, however, is a serious limitation since x = 0 is a singular point of Painlevé-III, and generic solutions of (1.1) will be singular at this point and behave like u(x) ≃ ax p , |Re(p)| < 1. More specifically, based on symbolic computation we expect the asymptotic expansion for solutions of (1.1) in the form

b kl x 2k+(2l-1)p + c kl x 2k+1+2lp as x → 0.

To tackle this issue, we develop a second approach that avoids series expansions and instead relies on the isomonodromy representation of the Painlevé transcendents. It was first discovered by Garnier [13] and further explicated by Jimbo and Miwa in [24] that Painlevé equations can be formulated as monodromy-preserving, or isomonodromic, deformations of corresponding 2 × 2 first-order systems of differential equations. This allows one to characterize solutions of a given Painlevé equation in terms of a 2 × 2 Riemann-Hilbert problem. Such a monodromy representation was obtained for rational solutions of Painlevé-III(D 6 ) in [5]. From this point of view, one can show that for fixed α, β ∈ C, the solutions of (1.1) are parametrized by triples (x 1 , x 2 , x 3 ) ∈ C 3 on the cubic surface, known as the monodromy manifold, given by x 1 x 2 x 3 + x 2 1 + x 2 2 + x 2 e -iπα/4e -iπβ/4 + x 1 1e -iπ(α+β)/4e -iπ(α+β)/4 = 0. (1.13) The exponential constants appearing as coefficients in (1.13) will appear in multiple equations, making it convenient to introduce the notation e 0 := e iπα/8 ̸ = 0 and e ∞ := ie -iπβ/8 ̸ = 0.

(1. 14) In Section 4, we reproduce the derivation of the cubic surface (1.13) carried out in [43] and connect the quantities x i with other invariant quantities that appear in the Riemann-Hilbert Problem 4.1 associated with PIII(D 6 ). In Section 5, we present an analogous parametrization of solutions of the D 8 degeneration (1.3) of PIII in terms of triples (y 1 , y 2 , y 3 ) ∈ C 3 appearing in the Riemann-Hilbert Problem 5.1 and satisfying y 1 y 2 y 3 + y 2 1 + y 2 2 + 1 = 0.

(1.15)

Away from its singular points, we parametrize points (x 1 , x 2 , x 3 ) on the cubic surface (1.13) using parameters e 1 , e 2 appearing naturally from the point of view of the Riemann-Hilbert problem. In fact, e 2 1 , e -2 1 are eigenvalues of a certain monodromy matrix for a circuit about the origin for a linear system, see (4.1). The parameter e 2 appears in the connection matrix for the same system, see (4.28). We call (e 1 , e 2 ) monodromy parameters. Definition 1.3 (see Section 4 for details). We say the monodromy parameters (e 1 , e 2 ) are generic if (i) e 4 1 ̸ = 1, (ii) e 1 e 2 ̸ = 0, (iii) e 2 1 ̸ = e ±2 ∞ and e 2 1 ̸ = e ±2 0 . Before moving on, we pause to make a few observations. where convergence is uniform on compact subsets of C \ Σ(y 1 , y 2 , y 3 ) slit along Arg(z) = ±π and Σ(y 1 , y 2 , y 3 ) is the union of all poles and zeros of z → U (z; y 1 , y 2 , y 3 ).

There is nothing fundamental about the exclusion of Arg(z) = ±π; in fact, the Riemann-Hilbert analysis below can be continued onto the universal cover of C \ {0} with a suitable Figure 2. Left: the solution u 10 (x) of (1.1) generated by ten iterations of (1.2) with seed u 0 (x) corresponding to monodromy data µ = 0.23 + 0.39i (see (1.16)), e 2 = -0.45 -0.96i and α = 40.5 + 0.63i, β = 40.98 + 0.59i. Right: the limiting solution U (z) of (1.3). The labeling of poles and zeros is the same as in Figure 1. Note that both u 10 (x) and U (z) are branched at the origin. extension of the set Σ(y 1 , y 2 , y 3 ). Similar observations apply to Proposition 1.5 and Theorem 1.7 below.

We illustrate Theorem 1.4 for solutions that are not single-valued near the origin in Figure 2. Note that while the point (x 1 , x 2 , x 3 ) ∈ C 3 on the monodromy manifold (1.13) only depends on the squares of e 1 , e 2 , the point (y 1 , y 2 , y 3 ) ∈ C 3 on the monodromy manifold (1.15) of the limiting solution U (z; y 1 , y 2 , y 3 ) of (1.3) has a sign ambiguity in the coordinates y 1 and y 2 . However, if either e 1 or e 2 changes sign, then the signs of y 1 and y 2 change together, and it turns out that the triples (y 1 , y 2 , y 3 ) and (-y 1 , -y 2 , y 3 ) both lie on the surface (1.15) together and correspond to the same solution of (1.3); see Remark 5.2 below. Similarly, there is no need for us to specify the sign of the square roots in (1.20)-(1.21) provided they are both taken to be the same. One might expect a similar ambiguity to arise from the replacement of e 2

1 → e -2 1 , since both are eigenvalues of the same matrix, but it turns out that (x 1 , x 2 , x 3 ) is invariant under this change provided e 2 is appropriately modified, and (y 1 , y 2 , y 3 ) remains invariant up to the sign ambiguity described above, see Remark 6.2 below.

The proof of Theorem 1.4 is given in Section 7, and relies on Riemann-Hilbert analysis. The idea of the proof is to use parametrices constructed out of confluent hypergeometric functions near zero and infinity to reduce the setup to a Riemann-Hilbert Problem 7.5 on the circle. After some additional transformations, the problem allows taking a large n limit which gives us a Riemann-Hilbert Problem 7.9 with a jump on the circle in terms of Bessel functions. Further transformations using parametrices constructed out of Bessel functions simplify the jump and we arrive at a Riemann-Hilbert Problem 5.1 for Painlevé-III(D 8 ). In Section 9.1, we transform this into another Riemann-Hilbert Problem 9.2 for (1.3) already known in the literature. It is worth pointing out that even in the case of rational solutions, the Painlevé-III(D 6 ) Riemann-Hilbert Problem 4.1 exhibits Stokes phenomenon near both singular points [5] and hence requires the use of confluent hypergeometric parametrices to desingularize the problem before passing to the limit.

While the formulae for y i are daunting, they drastically simplify in the case of the rational solutions, where u 0 has monodromy data parametrized by . Then, we have

One can check that with these choices of α, β, e 1 , and e 2 we have U (z; y 1 , y 2 , y 3 ) = U (z; m), cf. Theorems 1.1 and 1.4. By further specializing U (z; m) to m ∈ iR + Z, we arrive at highly symmetric solutions of PIII(D 8 ) which have appeared in various works in nonlinear optics [40] and as a limiting object of various families of solutions to the focusing nonlinear Schrödinger equation in different regimes [1,2]. Furthermore, these solutions can be identified with pure imaginary solutions of the radial reduction of sine-Gordon equation, see, e.g., [11,Chapter 13]. It is interesting that they are related to another limiting object appearing in the random matrix theory -the Bessel kernel determinant. The explicit relation is described in Corollary 3.4 below.

A consequence of the analysis in Section 7 below is a description of the behavior near the origin of solutions u(x) of (1.1) corresponding to generic monodromy parameters (e 1 , e 2 ). Proposition 1.5. Let u(x) be the solution of Painlevé-III(D 6 ) equation associated to µ, η ∈ C via the generic monodromy parameters given in

as x → 0 with |Arg(x)| < π where δ = min(1, 2 -4 Re(µ)) and ϵ = sgn(Re(µ)).

Proposition 1.5 appeared in [22, Theorem 3.2] and its derivation is given in [28] for an equivalent, degenerate Painlevé-V equation. We present its proof using a Riemann-Hilbert approach in Section 8, which follows the steps of the proof of Theorem 1.4. The case Re(µ) = 0 can be handled similarly, but we exclude it here because two distinct terms arise at the same leading order resulting in a more complicated formula. From this formula one can see that if Re(µ) = 0 the solution can exhibit sinusoidal oscillations with frequency diverging as x -foot_0 consistent with an essential singularity at the origin.

To apply Proposition 1.5 to the rational solutions (1.5), or more generally to the sequence of Bäcklund iterates starting from any seed solution of (1.1), requires knowledge of the corresponding sequence of monodromy data. This is the content of the following proposition, which we prove in Section 6.

Proposition 1.6. Let u 0 (x) be the solution of (1.1) with parameters (α, β) and monodromy data (µ, η) (see (1.16)) with -1 2 < Re(µ), Re(η) ≤ 1 2 . Then, the Bäcklund iterates u n (x) are parametrized by the following monodromy data e 2 1,n = e 2πiµn , e 2,n = e 2 , e 0,n = e iπ(α+4n)/8 , e ∞,n = ie -iπ(β+4n)/8 , where 1

and ϵ = sgn(Re(µ)).

One notable application of Propositions 1.5 and 1.6 is the case corresponding to the rational solutions of PIII(D 6 ) described above. In [8], the authors found a product formula for u n (0; m) (see (2.19) in Section 2). Applying Propositions 1.5 and 1.6 to this case yields the closed-form formula

Another observation is that the expression on the right-hand side of (1.24) in Proposition 1.5 evaluated at the n-dependent monodromy data from Proposition 1.6 and at argument x = z n has a finite limit along even and odd subsequences of n. The limiting expressions relate to the behavior of U (z; y 1 , y 2 , y 3 ), which we can take from the literature: 11,20,33]). Let U (z; y 1 , y 2 , y 3 ) be the solution of the Painlevé-III(D 8 ) equation (1.3) associated to (y 1 , y 2 , y 3 ) ∈ C 3 parametrized by generic monodromy parameters (e 1 , e 2 ) using formulae (1.20)- (1.22). Then, it holds that

We pause to note that coalescence between Painlevé equations has long been in the literature; a coalescence diagram of all six Painlevé equations already appeared in Okamoto's work [36], and was later expanded on in [34]. Later, a geometric interpretation of the coalescence was given in [6]. That being said, the above degenerations are carried out on the level of the differential equation, so that given a solution of a Painlevé equation, one does not have a characterization of the solution one arrives at under the coalescence procedure. Confluence on the level of the solutions of the differential equation has also appeared in the literature; one of the most interesting examples is the merging of regular singularities and corresponding creation of an irregular singularity. This process was studied in the works [16,29]. In the PhD thesis [18] the confluence was studied in more detail in the cases Painlevé VI → Painlevé V and Painlevé V → Painlevé III(D 6 ). In the works [26,27] the authors considered a transition from Painlevé II → Painlevé I that is different in nature.

In Section 2, we describe the coalescence map u → U in terms of initial conditions and prove Theorem 1.1 using Maclaurin series of these solutions. We apply it to Umemura polynomials in Section 3. In Sections 4 and 5, we describe the monodromy representations of PIII(D 6 ) and PIII(D 8 ), respectively. In Section 6, we explain the Schlesinger transformations underlying Gromak's Bäcklund transformation (1.2) and prove Proposition 1.6. In Section 7, we prove Theorem 1.4 by Riemann-Hilbert methods. We recycle the same methodology to prove Proposition 1.5 in Section 8. In Section 9.1, we perform a Fabry-typefoot_1 transformation to the Painlevé-III(D 8 ) Riemann-Hilbert problem naturally arising from our limit process to put it in more canonical form and justify its solvability.

2 Identifying the solution of the limiting Painlevé-III(D 8 ) equation using Maclaurin series

The Painlevé-III(D 6 ) equation (1.1) for u n (x; α, β) implies the following equivalent differential equation for U n (z; α, β) := u n (z/n; α, β):

where

Note that for arbitrary α ∈ C and β ∈ C fixed and n > 0 sufficiently large we have the following crude inequalities:

We construct solutions of (2.1) analytic at z = 0 as follows. First multiply (2.1) through by zU n (z) to obtain

We substitute into (2.4) a power series

and express all products through the Cauchy product formula. The left-hand side of (2.4) is then a formal power series in z, and assuming that υ 0 ̸ = 0, the coefficient of z 0 yields

the coefficient of z 1 yields

and for k ≥ 2, the coefficient of z k yields

We may omit the term with a = 0 from the first sum on the right-hand side. Using

along with the inequalities (2.3), the coefficients in the series (2.5) are subject to the inequalities

(2.9)

Now we define a sequence of positive numbers {Υ k } ∞ k=0 by taking Υ 0 > 0 arbitrary and setting

(2.12)

Following [21, Proposition 1.1.1, p. 261], we construct an algebraic equation formally satisfied by the power series

We first rewrite the generic k ≥ 2 equation in (2.12) in the equivalent form

Comparing with (2.13), this is the coefficient of z k in the power series expansion about z = 0 of the equation

More generally, since k ≥ 2 holds in (2.14), these relations are consistent also with the equation

We now pick the constants A, B, C so that (2.15) is also consistent with (2.10)-(2.11) and U (0) = Υ 0 in the series (2.13). Indeed, U (0) = Υ 0 is equivalent to the following equation obtained from the coefficient of z -1 in (2.15):

Then taking Υ 1 from (2.10), the constant term in (2.15) gives the equation

Finally, obtaining also Υ 2 from (2.11), the coefficient of z 1 in (2.15) is

The formal series (2.13) with the recurrence relations (2.10)-(2.12) is therefore consistent with the algebraic equation (rewriting (2.15) with the above expressions for A, B, C):

However, it is a straightforward application of the implicit function theorem to observe that equation (2.16) has a unique solution U = U (z) analytic at z = 0 with U (0) = Υ 0 > 0 (this condition guarantees that the root U = Υ 0 of the quadratic on the left-hand side of (2.16) is simple). This proves that the formal series (2.13) with coefficients determined from (2.10)-(2.12) has a positive radius of convergence for each given value Υ 0 > 0.

Theorem 2.1. Fix α ∈ C and β ∈ C and let {U n (z; α, β)} ∞ n=1 be a sequence of solutions of (2.1) that are analytic at the origin z = 0 and suppose that

Then there exists a radius ρ > 0 such that for all n sufficiently large U n (z; α, β) is analytic for |z| < ρ and such that U n (z; α, β) → U ∞ (z; α, β) as n → ∞ uniformly for |z| < ρ, where U (z) = U ∞ (z; α, β) is the unique solution of the Painlevé-III(D 8 ) equation (1.3) that is analytic at the origin with U ∞ (0; α, β) = υ ∞,0 .

Proof . Let {υ n,k } ∞ k=0 denote the power series coefficients of U n (z; α, β) as in (2.5). Define Υ 0 by Υ 0 > 2|υ ∞,0 | (say), and obtain the subsequent coefficients {Υ k } ∞ k=1 via (2.10)-(2.12). Comparing (2.9)-(2.12) then shows that for all n sufficiently large, |υ n,k | ≤ Υ k holds for all k = 0, 1, 2, . . .. For each fixed k = 0, 1, 2, . . . , the recurrence relations (2.6)-(2.8) together with the limit

) then follows by dominated convergence. That the limiting analytic function U ∞ (z; α, β) is a solution of (1.3) follows from passing to the limit in each term of (2.1) using (2.2). That this solution is the unique analytic solution of (1.3) with the specified value at z = 0 then follows from passing to the limit in the recurrence relations (2.6)- (2.8). ■

Now we apply this result to the rational solutions u n (x; m) of equation (1.1), corresponding to α = -β = 4m. To this end, we point out that in [8], the authors studied the Umemura polynomials s n (x; m) at x = 0, and we begin by recalling one of their results.

). Set y := m + 1 2 and write ϕ n (y) := s n (0; m). If n = 2k is even, then

It follows from two identities above that also

Using (1.5) and s n (0; m -1) = ϕ n m -1 2 one has

Therefore, from (2.17) one gets that

(2.20)

Similarly, from (2.18) one gets

(2.21)

Using the infinite product formulae (see [38, equations (4.22.1)-(4.22.2)])

we get the following result.

We then apply Theorem 2.1 and obtain the following Corollary, which completes the local proof of Theorem 1.1. . Then for all z in a neighborhood of the origin, we have

Note that the equation (1.3) is invariant under the Z 2 -Bäcklund transformation U (z) → -U (z) -1 , so the even/odd subsequences of rational solutions both tend to related solutions of the same equation.

In this section, we obtain asymptotic results about the Umemura polynomials s n (x; m) and, as a consequence, particular 2jk determinants (see (3.30) below).

3.1 Painlevé-III tau functions, the Toda lattice, and expressing s n (x; m) in terms of u n+1 (x; m)

As a first step, we would like to obtain an expression of the Umemura polynomials s n (x; m) in terms of the rational Painlevé-III solutions themselves. We follow closely the works [12,37]. We introduce the Hamiltonian H n ≡ H n (x; m) via the equation

where the momentum, p n ≡ p n (x; m), is given by We would like to fix the constant in this definition by choosing a path of integration going to x = ∞ in the sector |Arg(x)| < π. To this end, it was shown in [5] that the rational functions u n (x; m) behave at infinity as u n (x; m) = 1 + O x -1 . In fact, using this in the Painlevé-III(D 6 ) equation (1.1) with α = 4(n + m) and β = 4(nm) gives the more refined asymptotics

which, together with (3.2) implies that the right-hand side of (3.3) satisfies

Now, every pole x 0 ̸ = 0 of u n (x; m) is simple with residue ± 1 2 , and moreover directly from (1.1), we find that

Similarly, all zeros x 0 ̸ = 0 of u n (x; m) are simple, with u ′ n (x 0 ; m) = ±2, and again from (1.1) we have

These expansions can be differentiated with respect to x to obtain corresponding expansions of p n (x; m) via (3.2) and then of H n (x; m) via (3.1). These expansions show that the only possible singularities x 0 ̸ = 0 of the right-hand side of (3.3) are simple poles of residue 1 that occur at simple zeros of u n (x; m) with u ′ n (x 0 ; m) = -2. Furthermore, if m ̸ ∈ Z+ 1 2 , then u n (x; m) is analytic and nonzero at x = 0, and it follows that the right-hand side of (3.3) has a simple pole at the origin with residue -1 8 4(m-n+1) 2 -1 . Therefore, arbitrarily fixing an integration constant, the tau function τ n (x; m) then can be defined for m ̸ ∈ Z + 1 2 and |Arg(x)| < π by

where the power function denotes the principal branch, the path of integration lies in the sector |Arg(y)| < π avoiding all poles of the meromorphic integrand, and then the integral is independent of path modulo 2πi. It then follows from (3.4) that τ n (x; m) admits the expansion

and that τ n (x; m)x (4(m-n+1) 2 -1)/8 extends to a neighborhood of x = 0 as an analytic nonvanishing function. From the point of view of the function τ n (x; m) the recurrence (1.6), which defines the Umemura polynomials, is equivalent to the Toda equation. More precisely, if we define the function

then using Gromak's Bäcklund transformation (1.2) with u = u n (x; m), û = u n+1 (x; m), and α = 4(n + m), β = 4(nm), we can check that h n satisfies the identity

Similarly, using the inverse of Gromak's transformation (1.2):

, in which u(x) solves (1.1) and û(x) solves the same equation with parameters (α, β) replaced by (α + 4, β + 4), one can check the identity . We now choose the constants K n (m) so as to have C n (m) = 1. To this end, using the detailed asymptotics (3.6), one can check that the leading term of both sides of (3.10) as x → ∞ is proportional to x 2 and equating those coefficients under the assumption that C n (m) = 1 yields the equation

of which we choose a particular solution K n (m) = (2i) n 2 , which yields the expression τn (x; m) = (2i) n 2 e -x 2 x n 2 τ n (x; m).

(3.11)

Now if we put

τn+1 (x; m), then it follows from (3.10) with C n (m) = 1 that s n (x; m) satisfies the Umemura recurrence relation (1.6). Moreover, using

shows that the integrand in the exponent of τ 0 (x; m) and τ 1 (x; m) vanishes identically, from which it follows that the initial conditions (1.7) are satisfied as well. Since the recurrence relation and initial conditions together have a unique solution, using (3.5) and (3.11), the Umemura polynomials are given by We could equally well have used (3.13) to derive the Toda equation instead of (3.12), but it is nice to have two different formulae for Umemura polynomials.

The ratio s n (x; m)/s n (0; m) for large n and small x

The representation (3.13) can be combined with Theorem 1.1 to obtain a limiting formula for s n (x; m)/s n (0; m) as n → ∞ and x → 0 at related rates. First, we note that with the notation U n (z; m) := u n (z/n; m), from (3.2) we obtain

Next, note that by Theorem 2.1 we can differentiate the limit in Theorem 1.1 for z near the origin, and hence for small z and n even we have U n (z; m) → U (z; m) and U ′ n (z; m) → U ′ (z; m), while for n odd we have instead U n (z; m) → -U (z; m) -1 and U ′ n (z; m) → U (z; m) -2 U ′ (z; m). Therefore, we have the following limit:

where we take the plus sign for n even and the minus sign for n odd, and the convergence is uniform for |z| sufficiently small. It follows that if x = z/(n + 1) in (3.13), by the corresponding substitution y → y/(n + 1) with the limits being uniform for |z| sufficiently small. To reduce the right-hand side in each case to the corresponding formula presented in Theorem 1.2 we refer to Section 3.4 below.

3.3 Asymptotic behavior of s n (0; m) for large n

We now compute the large n asymptotics of s n (0; m) = ϕ n m + 1 2 . First we write the formula for ϕ n (y) from Lemma 2.2 in terms of Gamma functions

, n even,

Since we are interested in asymptotics for large n, we need to use the reflection formula for the Gamma function [38, equation (5.5.3)] in the denominator:

Next, we use the Gamma duplication formula [38, equation (5.5.5)] and get

Now we can rewrite ϕ n (y) in terms of the Barnes G-function:

as n → ∞, where A = e [38, equation (5.17.6)]. Recalling y = m + 1 2 , we complete the proof of the formulae (1.11)-(1.12).

We have already seen how the PIII(D 8 ) equation ( 1.3) can be obtained from the PIII(D 6 ) equation (1.1) by confluence. There exists another, less known relation between the two equations -namely, a quadratic transformation mapping the solutions of PIII(D 8 ) to solutions of PIII(D 6 ) with special parameter values. Moreover, for precisely this parameter choice the relevant PIII(D 6 ) admits a family of transcendental analytic solutions that can be expressed in terms of Fredholm determinants of the continuous Bessel kernel. Under quadratic transformations, they are mapped to solutions of PIII(D 8 ) analytic at z = 0. This allows one to give yet another characterization of the PIII(D 8 ) transcendent describing the large-order asymptotics of the rational PIII(D 6 ) solutions. Indeed, let U (z) be an arbitrary solution of the PIII(D 8 ) equation (1.3). It is then straightforward to check that the function σ(r) defined by

satisfies the σ-form of a particular PIII(D 6 ) equation, namely,

Indeed, letting ς(t) := σ(4t) + t transforms equation (3.17) to

The latter appears in [22, equation (3.13)] and [35, equation (E III ′ )]. These relate to (1.1) via the following transformations; letting

yields (a special case of) the so-called "prime" version of Painlevé-III

Next, letting t = x 2 and q(t) = xu(x) yields (1.1) with parameters α = 0 and β = 4. Combining the transformations U (z) → σ(r) → ς(t) → q(t) → u(x) yields an explicit formula for u(x) in terms of U (z):

.

Correcting for a typo, 4 this is equivalent to [2, equation (112)]. Note that if U (z) is a solution of (1.3) that is analytic at z = 0 with U (0) ̸ = 0, and hence also from

3 Observe that our definition of σ differs from that of [41] by a negative sign. 4 The relevant equation in [2] should be corrected to read

then (3.16) implies that σ(r) is analytic at r = 0 with σ(0) = 0, and, in fact,

Also, differentiating (3.16) and using (1.3) to eliminate U ′′ (z) yields the relation

which can be regarded as an algebraic equation expressing U (z) in terms of σ ′ (32iz). Conversely, any solution σ(r) of (3.17) different from an affine function ar + b can be mapped to a pair of solutions U (z) of PIII(D 8 ) related by the Z 2 Bäcklund transformation U (z) → -1/U (z) with the help of the formula (3.19). To see this, one first uses (3.17) to explicitly express σ(r) in terms of its derivatives and r, and then differentiates the resulting expression with respect to r. Each term of the resulting equation has a common factor of rσ ′′ (r). Hence if σ(r) is non-affine, one may cancel this factor, and then σ ′ (r), σ ′′ (r), and σ ′′′ (r) can be eliminated from the reduced equation using (3.19) and its derivatives. This implies that either U (z) 2 + 1 = 0 or U (z) is a solution of (1.3), and the latter admits precisely the constant solutions U (z) = ±i so we may conclude that any meromorphic function U (z) obtained from a non-affine solution of (3.17 The Bessel kernel can be equivalently written as

x m y n .

(3.21)

The first of these identities follows from the easily verified differentiation formula

whereas the second one is obtained by substituting into the integral expression the standard series representation of J 0 (•) [38, equation (10.2.2)]. Using (1.8) along with representation (3.21), then enables one to compute the traces of powers of K r in the form of a series in r. It yields

Expansions of such form are known for Fredholm determinants appearing in random matrix theory, see [31,Section 20.5]. Let us record explicitly the few first terms of (3.22):

which implies that the Bessel determinant solution of (3.17) guaranteed by Proposition 3.1 has the asymptotics

This expression is of course consistent with the differential equation (3.17).

On the other hand, if U (z) = U (z; m) is the particular solution of (1.3) relevant to Theorem 1.1, which for m ∈ C \ Z + 1 2 is analytic at the origin with

then according to (3.18), the corresponding solution of (3.17) analytic at the origin satisfies

(3.24)

Note that λ(m) is necessarily finite for m ∈ C \ Z + 1 2 and there are only two values it never takes for any m: λ(m) ̸ = 0, 1. Also, the coefficient of r 2 cannot vanish for any m ∈ C. Now we need the following result. Proposition 3.2. Let σ 1 (r) and σ 2 (r) denote two non-affine solutions of (3.17) both analytic at the origin and both satisfying σ j (r) = -

r) is a solution of (3.17) analytic at the origin with σ ′ (0) = - 1 4 λ, then it has a locally-convergent Taylor series

for some ρ > 0. Using this in the differential equation (3.17), from the coefficient of r 2 one obtains

whereas from the coefficient of

where on the right-hand side, s 1 := -1 4 λ. Now, (3.25) implies that either s 2 = 0 or s 2 =

since λ ̸ = 0, 1. We now use (3.27) as the base case for an inductive argument. Suppose (3.26) for the coefficient of r k+1 , we obtain

from which it follows that σ(r) = -1 4 λr exactly. This is a contradiction, because σ(r) is not affine. Therefore, s 2 ̸ = 0.

Taking s 2 = 1 16 λ(1λ) ̸ = 0 as necessary, we note that in (3.26) for k ≥ 3, s k appears only on the left-hand side with coefficient

while the right-hand side only involves s 1 , . . . , s k-1 . Therefore, all subsequent coefficients s k , k ≥ 3 are uniquely determined by the recurrence, implying that σ

, for which σ(r) defined by (3.20) is an affine function. See [41].

Since the analytic solutions with expansions given in (3.23) and (3.24) have the same leading term if λ = λ(m) ̸ = 0, 1, and neither solution is an affine function, they coincide for small |r|. Because the function U (z; m) is then determined up to the involution U → -U -1 by (3.19), we have proved the following result.

with λ(m) = 1/ 1 + e 2πim . In particular, the expansion of U (z; m) in powers of z can be read off from the series representation (3.22).

Furthermore, using (3.17

and we get Theorem 1.2.

On the other hand, the Umemura polynomials admit the following Wronskian determinant representation [25]:

where for parameter α ∈ C and index k ∈ Z, L

k (x) are generalized Laguerre polynomials for k ≥ 0, while L (α) k (x) = 0 for k < 0. Expressions like that on the right-hand side are called Wronskian Appell polynomials in [3]; similar formulae hold for rational solutions of other Painlevé equations as well. Wronskian determinants of generalized Laguerre polynomials were also studied in [7].

The generalized Laguerre polynomials admit the following integral representation [38, equation (18.10.8)]:

for |Arg(x)| < π (analytically continuable to x ∈ C) where ε > 0 is small enough so that the branch cut z ≤ 0 is outside the contour of integration. Making the transformation

We denote

Using this notation, we obtain

Similar "2jk" determinants have appeared in various works in the literature, see, e.g., [15]. Denoting

it immediately follows from (3.15), (3.14) that in the limit j → ∞,

To write the analog of Theorem 1.2 for D n (x; m) we need to compute the asymptotic behavior of n k=1 (2k -1)!!. We use [38, equation (5.4.2)] to get

Combining this with formulae (1.11), (1.12) and using (3.28), we get Following Jimbo and Miwa [24], we use the fact that each Painlevé equation can be recast as an isomonodromic deformation condition for a 2×2 system of linear ODEs with rational coefficients. The case of Painlevé-III (D 6 ) corresponds to the situation where the coefficient matrix for the equation in the spectral variable, λ, has exactly two poles on the Riemann sphere leading to irregular singularities at λ = 0 and λ = ∞, at each of which the leading term is diagonalizable. After some normalization, the differential equation can be written in the formfoot_2

where

In this case, the deformation equation is

where

In the expressions for Λ (6) (λ, x) and X(λ, x), Θ ∞ is a complex parameter and s = s(x), t = t(x), v = v(x), y = y(x). The equations (4.1) and (4.3) constitute an over-determined system with compatibility condition

where Λ (6) , X is the commutator. This boils down to the scalar equations

then it follows from (4.4) that

which can be seen to be equivalent to (1.1) by taking another x-derivative and using (4.4) again, after which the quantity

appears. However, from (4.4) it follows that I ′ (x) = 0, so denoting the constant value of I by Θ 0 , we arrive at (1.1) with parameters

The constants Θ 0 , Θ ∞ can be naturally interpreted on the level of the 2 × 2 system (4.1), which we now explore. For all the calculations that follow, we assume for simplicity that x > 0. The system (4.1) admits formal solutions near the singular pointsfoot_3

and

Here ∆ (6) (x) is an (invertible) eigenvector matrix of the coefficient of λ -2 in (4.2), so the leading term of ∆ (6) (x) -1 Λ (6) (λ, x)∆ (6) (x) at λ = 0 is diagonal. For k = 1, 2, 3, we define the Stokes sectors,

It follows from the classical theory of linear systems that there exist canonical solutions

k respectively and determined uniquely by the asymptotic condition

In these asymptotic conditions, the meaning of the power functions in (4.7) and (4.8) is determined from the range of Arg(λ) in the definition of S (ν)

k . The canonical solutions in consecutive Stokes sectors are related to one another by multiplication on the right with Stokes matrices, i.e.,

where for some Stokes multipliers s ∞,0

Likewise, by uniqueness and the different interpretation of the multi-valued powers in the formal solutions on the otherwise identical sectors S (∞,0) 1 and S

(∞,0) 3

, we have the identities

where, combining (1. k and since both canonical solutions solve (4.1) in the same domain, they must be related by multiplication on the right by a constant connection matrix, which we define using

The condition that the coefficients y, v, s, t in the matrix Λ (6) (λ, x) depend on x as a solution of (4.4) implies simultaneous solvability of (4.1) and (4.3), and the latter system implies that the Stokes matrices and connection matrices are, like Θ 0 and Θ ∞ , independent of x. We show below in Sections 4.3 and 4.6 that the four Stokes multipliers and the elements of the two connection matrices are determined from just two essential monodromy parameters that we denote by e 1 and e 2 .

Using the canonical solutions, we define the following sectionally-analytic function

. The jump contour L (6) for Ψ(λ, x) and definition of J Ψ (λ) when x > 0.

Then, it follows from the asymptotic conditions (4.9) and the relations (4.10)-(4.12) and (4.14)-(4.15) that Ψ solves the following 2 × 2 Riemann-Hilbert problem. Let λ p denote the branch of the power function analytic in C \ iR -with argument chosen so that

The notation reminds us that the branch cut of these functions is the contour carrying lower triangular Stokes matrices.

Riemann-Hilbert Problem 4.1. Fix generic monodromy parameters (e 1 , e 2 ) determining the Stokes and connection matrices, and x > 0. We seek a 2 × 2 matrix function λ → Ψ(λ, x) satisfying:

Analyticity: Ψ(λ, x) is analytic in C\L (6) , where L (6) = {|λ| = 1}∪ iR is the jump contour shown in Figure 3.

Jump condition: Ψ(λ, x) has continuous boundary values on L (6) \{0} from each component of C \ L (6) , which satisfy

, where J Ψ (λ) is as shown in Figure 3 and where the + (resp., -) subscript denotes a boundary value taken from the left (resp., right) of an arc of L (6) .

Normalization: Ψ satisfies the asymptotic conditions

and

where ∆ (6) (x) is a matrix determined from Ψ(λ, x) having unit determinant.

Observe that if Ψ solves Riemann-Hilbert Problem 4.1, then the following limit exists:

where ∆ (6) , Ξ (6) are as in (4.18), (4.19), respectively.

To study the direct monodromy problem and obtain the jump matrices given just the values of u and u ′ at an initial point x 0 , it is necessary to introduce artificial initial values of the auxiliary functions s, t, v, y at x 0 in way consistent with the definition (4.5) of u(x). Different consistent choices lead to different jump matrices, but the jump matrices determine the same function u(x) via (4.20). This symmetry is reflected at the level of Ψ(λ, x) by the conjugation Ψ(λ, x) → δ -σ 3 Ψ(λ, x)δ σ 3 for any δ ̸ = 0. Another symmetry that also leaves u(x) invariant but changes the jump matrices C ± 0∞ is multiplication of Ψ(λ, x) on the right for |λ| < 1 only by a unit-determinant diagonal matrix. Therefore, having obtained the jump matrices for the inverse monodromy problem via a direct monodromy calculation, after the fact we may introduce an arbitrary transformation of Ψ(λ, x) of the form

without changing u(x). This transformation modifies the Stokes matrices as follows:

and it modifies the connection matrices as

The cyclic products of the jump matrices for the inverse monodromy problem about the two non-singular self-intersection points of the jump contour λ = ±i read

We can use the second relation to explicitly write C + 0∞ in terms of two Stokes matrices and the other connection matrix:

This identity is an analog of [22, equation (3.17)]. Under the condition that det C - 0∞ = 1, we immediately get that det C + 0∞ = 1. Furthermore, using (4.25) we eliminate C + 0∞ from the first equation of (4.24) to obtain the identity

In other words, (S ∞ 1 ) -1 e -2σ 3 ∞ (S ∞ 2 ) -1 and S 0 1 -1 e 2σ 3 0 S 0 2 -1 are similar unit-determinant matrices. Note that this is merely reflective of the fact that both products are monodromy matrices, possibly expressed in terms of different bases of fundamental solutions, for a simple circuit about the origin for solutions of the system (4.1). Let us assume that they have distinct eigenvalues that we will denote e ±2 1 . Then, both products are diagonalizable, so there exist unit-determinant eigenvector matrices E ∞ and E 0 such that

To specify the eigenvector matrices E ∞ , E 0 uniquely, we agree that their (2,2) entries are both equal to 1. Using (4.27) in (4.26) gives a homogeneous linear equation on C - 0∞ that can be written in commutator form as

1 can be written in the form

Under the assumption e 4 1 ̸ = 1 we already invoked to obtain diagonalizability, g ̸ = 0 so the commutator equation implies that (E ∞ ) -1 C - 0∞ E 0 is a diagonal unit-determinant matrix that we may write in the form e σ 3

2 . Thus we have the identity

Taking the trace of (4.26), we get

It is clear that one can solve for the products s ∞ 1 s ∞ 2 and s 0 1 s 0 2 in terms of e 2 1 , e 2 ∞ and e 2 1 , e 2 0 respectively. Using the transformation (4.22), we can take a particular solution of this relation and hence obtain the Stokes multipliers:

With the Stokes matrices specified in this way, the eigenvector matrices E ∞ and E 0 are uniquely specified as mentioned earlier by taking the (2, 2) entry to be 1 in each case, which yields

. The analogue of the contour L (6) in Figure 3 when |Arg(x)| ̸ = 0.

After making such choices, we obtain the formulae (1.17)- (1.19). At this point, it can be directly checked that our choices are consistent with the full equation (4.26) with C - 0∞ given by (4.28). One can think of fixing the (2, 2) entry in the following way: the eigenvector matrices E ∞ and E 0 represent "internal degrees of freedom" that have an additional symmetry, namely, arbitrary scalings of the eigenvectors that preserve determinants. In other words, while (4.21) induces a conjugation symmetry on the eigenvector matrices, there is an additional symmetry for each involving multiplication on the right by an arbitrary unit-determinant diagonal matrix. Thus, the matrices E ∞ and E 0 undergo the transformations

for some arbitrary nonzero quantities ϵ ∞ , ϵ 0 . Note that these transformations along with (4.23) and e

By contrast, e 2 1 → e 2 1 := e 2 1 is a symmetry invariant.

Remark 4.3. In the case where one is interested in values of x ∈ C with |Arg(x)| < π, the analogue of Figure 3 is shown in Figure 4, where the nonsingular self-intersection points are at λ = ±ie ±i Arg(x) (independent ± signs). The angles of the rays in the contour L (6) are chosen so that iλx ∈ R on the rays extending to λ = ∞, and iλ -1 x ∈ R on the rays extending to λ = 0. Similar to Section 4.2, one can formulate a Riemann-Hilbert problem for a sectionally analytic function λ → Ψ(λ, x) off of the contour L (6) and one finds four connection matrices instead of two, denoted C 1 through C 4 , defined on corresponding arcs of the unit circle as shown in Figure 4. These satisfy cyclic conditions similar to (4.24), namely,

Eliminating all but C 3 from the above identities yields the analog of (4.26), namely,

Reasoning similar to that of Section 4.3 yields

which, in turn, yields

In this setting, we must adjust our choice of the branch λ → arg (λ), and we choose a branch which satisfies (cf. (4.16) when Arg(x) = 0)

A concrete branch cut is chosen later, see Remark 7.1 below.

one obtains

With this choice of γ, δ and E ∞ , E 0 chosen as in (4.31), (4.32) (that is, we insist that the (2, 2) entry of E ∞ , E 0 is 1 by setting ϵ 0 = ϵ ∞ = 1), the connection matrices are

Remark 4.4. The above gauge is only needed to match our setup with that of [5]; in the sequel we will be working with γ = δ = 1. Formula (4.28) then implies

This is important to note when, for example, one tries to verify that (8.8) below reduces to (1.25).

Before beginning to study the large n behavior of u n , we must first establish a similar monodromy representation of the limiting solution of Painlevé-III(D 8 ), which we do in Section 5 below.

It is known that the monodromy manifold for Painlevé-III(D 6 ) can be given by a cubic equation (see, e.g., [43]), which can be recovered from our point of view as follows. Denote

Then, the inverse of the cyclic relation (4.26) allows us to solve for s ∞ 1 , s ∞ 2 in terms of parameters m i , ℓ i , and imposes the constraint

Hence, we are left with these eight parameters subject to the constraint (4.35) and the unitdeterminant conditions

We may define coordinates which are invariant under the transformation (4.21):

Equations (4.35), (4.36) imply e 2 ∞ = e -2 0 I 1 -I 2 + I 3 -I 4 (I 1 -1),

We eliminate I 3 and get

Introducing new variables

yields the following equation, which defines the monodromy manifold for the problem

This matches (1.13) upon using (4.6) and (4.13). Using (4.37), we obtain formulae (1.17)- (1.19).

To find the singularities of (4.38), we adjoin to (4.38) the three equations obtained by setting to zero the components of the gradient vector of the left-hand side of (4.38) with respect to (x 1 , x 2 , x 3 ). There is therefore at most one singularity:

)

In particular, if neither e -2 0 = e 2 ∞ nor e 2 0 = e 2 ∞ , then the monodromy manifold is a smooth curve with no singular points. Notice that we can use (x 1 , x 2 ) as parameters for the generic collection of points on monodromy manifold (4.38) for which x 1 x 2 ̸ = 0, because x 3 can be explicitly expressed in terms of the other coordinates. The points satisfying (4.38) with x 1 = 0 form a 1-dimensional variety consisting in general of two distinct lines:

∞ , the two lines coincide and pass through the critical point (4.39) of (4.38). Likewise there are generally two lines on (4.38) along which x 2 = 0 each parametrized by x 3 ∈ C:

and if e 2 0 = e 2 ∞ , the two lines again coincide and pass through the critical point (4.40) of (4.38).

5 General monodromy data: Painlevé-III(D 8 )

5.1 Lax pair for Painlevé-III(D 8 )

The Painlevé-III(D 8 ) equation ( 1.3) can also be formulated as an isomonodromic deformation of a linear system. In this case we need two ramified irregular singularities at λ = 0 and λ = ∞, i.e., we consider the system

where

and functions U (z), V (z), W (z), X(z) satisfy the identities

Note the characteristic feature that the leading terms of Λ (8) (λ, z) and of Z(λ, x) at the singular points λ = 0, ∞ are singular and nondiagonalizable matrices.

Since Ω(λ, z) is a simultaneous fundamental solution matrix of the Lax system (5.1)-(5.2), the zero-curvature compatibility condition for that system is therefore satisfied:

Equating to zero the coefficients of different powers of λ on the left-hand side gives a first-order system of four differential equations on the four functions U (z), V (z), W (z), and X(z):

It is possible to express the functions W (z), X(z), and V (z) in terms of U (z) and U ′ (z) using (5.3), (5.4), and (5.5), but since we do not use these formulae, we do not present them here. Using (5.5) to repeatedly eliminate all derivatives, it is straightforward to obtain the following identity:

Of course the right-hand side vanishes as a result of the identity (5.3). Hence U (z) is a solution of (1.3), the Painlevé-III(D 8 ) equation.

For all the calculations that follow, we assume for simplicity that z > 0. The system (5.1) admits formal solutions near the singular points

and

where

and the square roots denote principal branches. The function ∆ (8) (z) satisfies the identity

and hence the solution U (z) can be expressed as

(5.9)

For k = 1, 2, we define the Stokes sectors,

It follows from the classical theory of linear systems that there exist canonical solutions

k determined uniquely by the asymptotic condition

k , ν ∈ {0, ∞}.

(5.10)

The canonical solutions in consecutive Stokes sectors at λ = 0, ∞ are related to one another by multiplications on the right with Stokes matrices, i.e.,

)

)

)

where k and since both canonical solutions solve (4.1) in the same domain, they must be related by multiplication on the right by a constant connection matrix, which we define using

(5.16)

In a fashion similar to Section 4.2, we now formulate a 2 × 2 Riemann-Hilbert problem for a sectionally-analytic function Ω defined by

0 (λ, z), |λ| < 1 and -

Then, it follows from the asymptotic conditions (5.10) and the relations (5.11)-(5.14) and (5.16) that Ω solves the following 2 × 2 Riemann-Hilbert problem.

Riemann-Hilbert Problem 5.1. Fix monodromy data t 0 0 , t ∞ 1 and z > 0. We seek a 2 × 2 matrix function λ → Ω(λ, z) satisfying: Analyticity: Ω(λ, z) is analytic in C \ L (8) , where L (8) = {|λ| = 1} ∪ iR -is the jump contour shown in Figure 5.

Jump condition: Ω(λ, z) has continuous boundary values on L (8) \{0} from each component of C \ L (8) , which satisfy

where J Ω (λ) is as shown in Figure 5.

Normalization: Ω(λ, z) satisfies the asymptotic conditions

and

where ∆ (8) (z) is a matrix determined from Ω(λ, z) having unit determinant.

Solvability of Riemann-Hilbert Problem 5.1 is discussed in Section 9.1.

. The jump contour L (8) and definition of J Ω (λ) when z > 0.

Since the jump matrices depend on neither λ nor z, the matrices

are both analytic functions of λ in the domain C \ {0}. We determine these analytic functions by computing sufficiently many terms in their asymptotic expansions as λ → ∞ and λ → 0 using (5.6)-(5.7). We will use the identities

and

Using (5.18) and (5.6) gives, in the limit λ → ∞, the expansions

21 (z) 4iz(Ξ

21 (z) 4iz(Ξ

Actually, we can also go to higher order and compute the coefficient of λ -2 in the matrix element Λ 21 (λ, z), in the limit λ → ∞:

Likewise, using (5.19) and (5.7) gives that as λ → 0

Applying Liouville's theorem yields the exact expressions

.

Using the notation (5.9) and noting the structure of the coefficients of the different powers of λ, it is convenient to reparametrize the coefficients as follows:

and

-X(z) .

The quantities U (z), V (z), W (z), and X(z) are not independent; comparing the 21-element of the coefficient of λ -1 in the expansion of ∆ (8) (z) -1 Λ (8) (λ, z)∆ (8) (z) computed using (5.21) and (5.23) gives the identity (5.3). At the same time from formula (5.22) we get identity (5.4). Since (5.17) holds for the same matrix function Ω(λ, z), the latter satisfies the equations of a compatible Lax system ∂Ω ∂λ (λ, z) = Λ (8) (λ, z)Ω(λ, z) and ∂Ω ∂z (λ, z) = Z(λ, z)Ω(λ, z), (5.25) which coincides with the system (5.1)-(5.2).

Introducing notation for the connection matrix elements

we have the cyclic relation around the unique nonsingular point of self-intersection of L (8)

1 , the condition det(C 0∞ ) = 1 implies that the coordinates (y 1 , y 2 , y 3 ) are related by the cubic equation (1.15). Remark 5.2. If the solution Ω(λ, z) is multiplied by the scalar -1 for |λ| < 1 and left unchanged for |λ| > 1, then the elements of the connection matrix C 0∞ change sign while the Stokes multiplier t ∞ 1 is invariant. Therefore, this transformation changes (y 1 , y 2 , y 3 ) to (-y 1 , -y 2 , y 3 ), yielding a different point on the cubic (1.15). The matrix coefficient ∆ (8) (z) also changes sign, however ∆ (8) 21 (z) 2 is invariant, so the solution U (z) of the Painlevé-III(D 8 ) equation (1.3) is the same for both points.

6 Schlesinger transformation and proof of Proposition 1.6

Fix generic monodromy parameters (e 1 , e 2 ). In view of the parametrization of the Stokes multipliers in (4.30) and the eigenvector matrices in (4.31), (4.32), this data determines from Riemann-Hilbert Problem 4.1 a matrix Ψ(λ, x) which is meromorphic in x and satisfies asymptotic conditions (4.17) and (4.18), which we write in the formfoot_4

Define the matrices σ + = 1 0 0 0 and σ -= 0 0 0 1 .

Following [5], assuming the (1, 1) entry of Ψ 0 0 (x), denoted Ψ 0 0,11 (x), is not identically zero, we consider the Schlesinger transformation

where

Since λ ±1/2 has its branch cut along part of the curve L (6) , we see that Ψ(λ, x) is analytic in C \ L (6) and, by direct calculation, has the jumps on L (6) summarized by Figure 3, with the exception of the sign changes

Furthermore, one can verify using the definition of Ψ(λ, x) that

where

Similarly, one can check that

where

The transformation Ψ → Ψ is invertible so long as Ψ0 0,22 (x) does not identically vanish, and its inverse is given by

where

.

It follows that Ψ satisfies conditions similar to (6.1) and (6.2) as λ approaches ∞, 0, respectively. That these operations are inverses of one another is the content of [5, Lemma 1].

In this way, starting with Ψ and iterating the map Ψ → Ψ (assuming Ψ 0 0,11 (x), Ψ 0 0,22 (x) do not identically vanish after each step), we may define the nth iterate of this Schlesinger transformation, which we denote Ψ n . This matrix, if it exists, satisfies the following Riemann-Hilbert problem.

Riemann-Hilbert Problem 6.1. Fix generic monodromy parameters (e 1 , e 2 ), n ∈ Z, and x > 0. We seek a 2 × 2 matrix function λ → Ψ(λ, x) satisfying: (6) , where L (6) = {|λ| = 1} ∪ iR is the jump contour shown in Figure 3.

Jump condition: Ψ n (λ, x) has continuous boundary values on L (6) \ {0} from each component of C \ L (6) , which satisfy

where J Ψn (λ) is as shown in Figure 3 but with the modification

Normalization: Ψ n (λ, x) satisfies the asymptotic conditions

and

where ∆

n (x) is a matrix determined from Ψ n (λ, x) having unit determinant.

That Ψ n solves the above Riemann-Hilbert problem implies the existence of the limit

It follows that the function

n,12 (x) ∆

It was shown in [5,Lemma 2] that if for some n ∈ Z the inverse monodromy problem is solvable for a given x ∈ D, where D is a domain in C \ {0}, then Ψ n satisfies the Lax pair

where potentials s, t, u, v, y all depend on x and n. Furthermore, in this domain, the functions Ψ 0 0,11 (x), Ψ 0 0,22 (x) extracted from Ψ n (λ, x) are not identically zero. 8One can check that if a solution to Riemann-Hilbert Problem 6.1 exists, it must be unique, and we attempt to identify this solution as a solution of Riemann-Hilbert Problem 4.1 with possibly different monodromy data. The diagonal elements of S 0 2 e -2σ 3 0 , S ∞ 2 e 2σ 3 ∞ alternate signs which implies the change

Furthermore, in view of (4.30), we can write

) n e 2 0 -1 (-1) n e -2 0 . Combining the above with the fact that C ± 0∞ remain invariant under the iterated Schlesinger transformations implies the change in monodromy data Since e 1 , e 2 are assumed to be nonvanishing, we may write them in the form (1.16) for some µ, η ∈ C with -1 < Re(µ), Re(η) ≤ 1. Moreover, since the transformations e 1 → -e 1 , e 2 → -e 2 preserve the monodromy data, we can assume -1 2 < Re(η) ≤ 1 2 and -1 2 < Re(µ) ≤ 1 2 . Equation (6.7) implies in turn that η does not depend on n ∈ Z, while µ is replaced with

This proves Proposition 1.6. We end this section with two important remarks.

Remark 6.2. It was noted in the introduction that one could restrict 0 < Re(µ n ) ≤ 1/2, in which case, the above iterations interchange the roles of e 2 1 , e -2 1 and we have to perform the transformation µ → -µ, which corresponds to the replacements Therefore, we alternatively can write the monodromy data for the Schlesinger transformation as

Furthermore, one can check that (x 1 , x 2 , x 3 ) in (1.17)-(1.19) remain invariant under the map described in (6.9), whereas (y 1 , y 2 , y 3 ) → (±y 1 , ±y 2 , y 3 ) where the sign depends on the choice of the square root in (6.9) and (7.52) below. In both cases, the corresponding solution of (1.3) remains invariant, see Remark 5.2.

Remark 6.3. Moving forward, we will slightly abuse notation by suppressing the n-dependence in the parameters e ∞ = e πi(Θ∞-n)/2 , e 0 = e πi(Θ 0 +n)/2 , e 1 = e πiµn , e 2 = e πiη . (6.10)

7 Asymptotics for large n and small x and proof of Theorem 1.4

Let (e 1 , e 2 ) be generic monodromy parameters, see Definition 1.3. At this point, we can see more clearly the meaning of the genericity conditions formulated there:

(i) e 4 1 ̸ = 1; this is to guarantee diagonalizability in (4.27), (ii) e 1 e 2 ̸ = 0; this is to guarantee the unit-determinant condition in (4.27) and (4.28), (iii) e 2 1 ̸ = e ±2 ∞ and e 2 1 ̸ = e ±2 0 ; this, in particular, implies that the Stokes multipliers (4.30) are nonvanishing. Riemann-Hilbert Problem 7.2. Fix generic monodromy parameters (e 1 , e 2 ) determining the Stokes and connection matrices, n ∈ Z, and x > 0. We seek a 2 × 2 matrix function λ → Φ(∞) n (λ, x) satisfying: ) , where

is the jump contour shown in Figure 8.

where J Φ(∞) n (λ) is as shown in Figure 8 and where the + (resp., -) subscript denotes a boundary value taken from the left (resp., right) of an arc of Γ (∞) .

Φ(∞) n (λ, x) satisfies the asymptotic conditions

where A n (x) has zero trace and B n (x) has unit determinant.

It is easy to see that Φ(∞) n (λ, x) necessarily has unit determinant. Furthermore, note that the jump matrix being e -2σ 3 1 across the arc terminating at the origin implies e 1 = e πiµn , which is consistent with (6.10). 0

It follows from assuming differentiability of the asymptotics in (7.1)-( 7

and

Since the quantity on the left-hand side of (7.3) and (7.4) is otherwise an analytic function of λ, it follows from Liouville's Theorem that

Noting that Tr B n (x)σ 3 B n (x) -1 = 0 and det B n (x)σ 3 B n (x) -1 = -1, we may write

and use this form in (7.5) to write a coupled scalar system of differential equations satisfied by the elements ϕ 1 (λ, x) and ϕ 2 (λ, x) of the first and second rows, respectively, of any column of Φ(∞) n (λ, x):

)

Before beginning to solve this system, observe that equating the coefficients of λ -1 in (7.3) and (7.4) yields the identity

Since µ n and n are constants, this equation implies that a n (x) is independent of x, so we will simply write a n going forward. Now, solving for ϕ 1 (λ, x) in (7.8) and eliminating it from (7.7) yields (assuming c n (x) ̸ = 0 and using b

It is easy to see that the first-order derivative term is removed by the substitution ϕ 2 (λ, x) = λ -1/2 w(λ, x). Indeed, w(λ, x) satisfies

Finally, the explicit x-dependence in the coefficients can be removed by setting Z := ixλ and writing w(λ, x) = W (Z). Note that the notation W (Z) here is not related to W (z) appearing in Section 5. In this case, W (Z) satisfies the ordinary differential equation

which is Whittaker's equation (see [38,Chapter 13]) with parameter

Given ϕ 2 (λ, x) = λ -1/2 W (Z) for Z = ixλ, and a solution W (Z) of (7.11), it follows from (7.8) that the corresponding first-row entry is

A fundamental pair of solutions of (7.11) is given by W (Z) = W ±κn,µn (±Z), arg(±Z) ∈ (-π, π).

If we take the particular solution ϕ 2 (λ, x) = λ -1/2 W κn,µn (Z), then using the identity

(see [38, equation (13.15.23)]) in (7.13) gives

Likewise, if we take the particular solution ϕ 2 (λ, x) = λ -1/2 W -κn,µn (-Z), then using the identity

(see [38, equation (13.15.26)]) in (7.13) yields

Taking linear combinations with coefficients depending generally on the parameter x, the general solution matrix for the system (7.5) can be written in the form

where

) is a specific fundamental solution matrix of (7.5) constructed from

and

in which κ = κ n and µ n are given by (7.12) and K(x) is a matrix of free coefficients.

Going back to (7.1)-( 7.2) and now assuming that the asymptotics are differentiable with respect to x,

So, applying Liouville's theorem yields

and it follows from (7.9

where µ n , a n are independent of λ, x. To determine the x-dependence of c n (x), we use (7.18) to assemble (7.5) (using also (7.9) and (7.12)) and (7.17) to give the Lax system

) is a simultaneous fundamental solution matrix for these equations, the Lax system is compatible. The compatibility condition reads Λx (λ, x) -Xλ (λ, x) + Λ(λ, x), X(λ, x) = 0, which is equivalent to

for some constant γ n ̸ = 0. Thus, the coefficient c n (x) is determined up to the choice of the constant γ n . Note also that the coefficient matrices Λ(λ, x) and X(λ, x) are obviously related by the simple identity

Since the fundamental matrix Φ (∞) n (λ, x) defined by (7.14) satisfies (7.19), then so does

n (λ, x)K(x), and K(x) must now be chosen so that (7.20) is satisfied. Substituting into (7.20), we obtain an ordinary differential equation on K(x):

n (λ, x) written in (7.14), we have both

where we also used (7.19). Using this in (7.23) along with the explicit definition (7.15) of H(λ, x) and the identities (7.21) and (7.22) gives

Therefore, the x-dependence of the matrix K(x) is explicitly given by

where K is now independent of both λ and x. However, as the domain of analyticity of Φ(∞) n (λ, x) in the λ-plane consists of three disjoint regions, we expect to have to specify a different matrix K for each. Note also that the constant γ n remains to be determined.

n (λ, x) on the two regions with |λ| > 2

To fully specify the parametrix Φ(∞) n (λ, x) for |λ| > 2, we concretely take the jump contours for |λ| > 2 to lie along the real axis in the Z-plane, corresponding to R + and R -, respectively.

Thus, the part of the domain of analyticity of Φ(∞) n (λ, x) with |λ| > 2 has two components, corresponding to the upper and lower half Z-planes. To properly define Φ(∞) n (λ, x) in these two exterior domains, we firstly take the matrix factor H(λ, x) defined in (7.15) in the precise form

Then, we assume different constant matrices K = K ± n in the two domains by writing the parametrix for |λ| > 2 and ± Im(Z) > 0 as

We now express the matrices K ± n in terms of the remaining constants µ n and γ n by enforcing the asymptotic condition (7.1) in each of the two sectors with |λ| > 2. According to [38, equation (13.19.3)],

holds for each δ > 0. Hence also

Under the condition given on arg(Z), we have

Z) > 0 (i.e., 0 < Arg(Z) < π), e -iπκ , Im(Z) < 0 (i.e., -π < Arg(Z) < 0).

To calculate Z ±κ , we recall Z = ixλ and use [5, equation (49)]:

Next,

and hence, for any x ∈ C \ {0} such that |Arg(x)| < π, Z ±κ = x ±κ λ ±κ e ±iπκ/2 , Im(Z) > 0, e ∓3iπκ/2 , Im(Z) < 0.

Therefore, for λ large such that Im(Z) > 0,

2 )σ 3 e -ixλσ 3 /2 , so choosing K + n so that (7.25) is consistent with (7.1) in the sector Im(Z) > 0 requires that K + n is an off-diagonal matrix, namely,

Similarly, for λ large such that Im(Z) < 0,

2 )σ 3 e -ixλσ 3 /2 , so consistency of (7.25) with (7.1) in the sector Im(Z) < 0 requires

Some additional useful information can be gleaned by enforcing on Φ(∞) n (λ, x) the jump conditions for |λ| > 2. The jump rays are illustrated in the Z-plane with their orientations in Figure 9. The Whittaker function W κ,µn (Z) can be viewed as an analytic function on the cut plane |Arg(Z)| < π, and it follows from the connection formula [38, equation (13.14.13)] that the boundary values on the negative real axis are related by

Note that the denominators in the second term on the right-hand side of (7.26) are finite due to condition (iii) in the definition of generic data; see the beginning of Section 7. Indeed, it follows from (6.10) and (7.12) that

On Z ∈ R -the left (+) and right (-) boundary values correspond to limits from Im(Z) < 0 and Im(Z) > 0, respectively. Therefore, the second column of W(Z; κ, µ n ) is continuous across R -, and from (7.26) (replacing Z with -Z),

Here, on the third line we used the definition (7.16) of α κ,µn and the factorial identity

Requiring that this matches with the corresponding jump condition in Figure 8 gives the condition

For Z ∈ R + the ± boundary values correspond to the limit from Im(Z) ≷ 0. Therefore, now the first column of W(Z; κ, µ n ) is continuous across R + , and from (7.26),

Again here, the finiteness of the denominators is guaranteed by condition (iii) at the beginning of Section 7. Since H(λ, x) changes sign across iR -, we get that Φ(∞

The last two equalities follow by a direct calculation using the definitions of s ∞ 2 , κ n , e 1 , and e ∞ in (4.30), (7.12), and (6.10), respectively, along with the expression

and the classical identity

The parametrix Φ(∞) n (λ, x) in the region |λ| < We use the identity [38, equation (13.14.33)] to express the elements of W(Z; κ, µ) in terms of the alternative basis of solutions M -κ,±µ (-Z) of Whittaker's equation with parameters (κ, µ) that form a numerically satisfactory pair in a neighborhood of the origin and that are analytic for Arg(-Z) ∈ (-π, π). Moreover, these functions are the Maclaurin series associated with the regular singular point at Z = 0, so they have the property that

where the power function denotes the principal branch and where the error term represents an analytic function of Z vanishing at the origin. To deal with the first column of W(Z; κ, µ) we also use the corresponding identity M κ,µ (Z) = e ±iπ( 1 2 +µ) M -κ,µ (-Z) which holds for ± Im(Z) > 0 (see also [38, equation (13.14.10)]). Using the above identities, and under the condition 2µ ̸ ∈ Z (which follows from the condition (i) at the beginning of Section 7 in our case), we can write the elements of W(Z; κ, µ) in the form

and

These expressions can be usefully combined into a matrix identity:

where

and

To define the parametrix Φ(∞) n (λ, x) for |λ| < 2, we first introduce a constant matrix by

The equality of these two expressions can be seen as follows. First, combining (7.25) and (7.30), and using the fact that the matrix M(ixλ; κ, µ n ) is analytic in a neighborhood of λ = 2i, the jump condition for Φ(∞) n (λ, x) across the positive imaginary axis for |λ| > 2 shown in Figure 8 implies the identity G + κn,µn K + n S ∞ 1 = G - κn,µn K - n , which yields the desired equality. Then, we set

It is straightforward to then check that, regardless of the choice of µ n , the matrix J n defined by (7.32) is diagonal. Comparing (7.25) and (7.33) shows that the jump conditions for Φ(∞) n (λ, x) across the arcs of the circle |λ| = 2 shown in Figure 8 are satisfied. Using (7.29) then proves that Φ(∞) n (λ, x) satisfies the simple jump condition across the negative imaginary axis with |λ| < 2 shown in Figure 8 and that an expansion of the form shown in (7.2) holds. To check that the matrix J n is diagonal and arrive at its final form below, we use the identity (7.28) to get

The result is that the diagonal matrix J n from (7.32) is given by

Analyticity:

, where

is the jump contour shown in Figure 10.

Jump condition: Φ(0) n (λ, x) has continuous boundary values on Γ (0) \ {0} from each component of C \ Γ (0) , which satisfy

where J Φ(0) n (λ) is as shown in Figure 10 and where the + (resp., -) subscript denotes a boundary value taken from the left (resp., right) of an arc of Γ (0) .

Φ(0) n (λ, x) satisfies the asymptotic conditions

where O(1) refers to a function analytic and bounded in a neighborhood of λ = ∞ and

as λ (7.37)

We can write down the unique solution Φ(0) n (λ, x) explicitly in terms of the parametrix Φ(∞) n (λ, x) obtained in Section 7.2, but taken with the index 1n instead of n and Θ ∞ replaced by Θ 0 . If we indicate the dependence of Φ(∞) n (λ, x) and Φ(0) n (λ, x) on Θ ∞ and Θ 0 respectively with the notation

x, Θ 0 ), then we have the following. Proposition 7.4. Fix Θ ∞ , Θ 0 ∈ C and generic monodromy parameters (e 1 , e 2 ). Then

where

Proof . The mapping λ → -λ -1 takes the contour Γ (0) onto the contour Γ (∞) up to the reversal of orientation of certain arcs, and swaps the circles centered at the origin of radii 1 2 and 2. Therefore, the domain of analyticity of Φ(0) n (λ, x, Θ 0 ) is as desired. Under the map n → -n, Θ ∞ → Θ 0 , the exponentials defined in (6.10) satisfy e 2 0 → e 2 ∞ , whereas µ n = µ -n since this quantity depends only on the parity of n. This implies that the Stokes matrices defined in (4.11)-(5.15) satisfy the corresponding identities

Comparing Figures 8 and 10 then shows that the function defined by (7.38) satisfied the required jump conditions across the imaginary axis for |λ| < 1 2 . Likewise, the jump condition on the negative imaginary axis for |λ| > 1 2 is easily verified due to the identity valid for any a ̸ = 0:

Finally, the fact that Φ(0) n (λ, x, Θ 0 ) defined by (7.38) satisfies the required jump conditions across the circle |λ| = 1 2 follows from the corresponding jump conditions for Φ(∞) 1-n (λ, x, Θ 0 ) for |λ| = 2 and the identities

among the matrices E ∞ , E 0 defined in (4.31)-(4.32), which hold for the value of β n indicated in (7.39). It only remains to verify the asymptotics in (7.36)-(7.37). However, these follow from the corresponding formulae in (7.1)-(7.2) with the help of the identity

which holds for all λ not on the negative imaginary axis. Since the matrix function Φ(0) n (λ, x, Θ 0 ) defined by (7.38)-(7.39) satisfies all the required Riemann-Hilbert conditions, and there is at most one solution of those conditions, as is easily confirmed by a Liouville argument, the proof is finished. ■

The parametrix for Φ n (λ, x) is by definition the following matrix function:

This matrix function satisfies exactly the same jump conditions in the domains |λ| > 1 and |λ| < 1 as does Φ n (λ, x) itself, and it is also consistent with the asymptotics given in (6.3)-(6.4) (note that Ψ n (λ, x) = Φ n (λ, x) for |λ| sufficiently large or small). The parametrix has unit determinant, so the matrix quotient

is an analytic function of λ except possibly on the jump contour Γ shown in Figure 6 and on the unit circle, where there is a discontinuity in the definition of Φn (λ, x). However, since the jumps of Φn (λ, x) and Φ n (λ, x) agree on Γ, a Morera argument shows that Q n (λ, x) is actually analytic both for |λ| > 1 and for 0 < |λ| < 1. The asymptotic behavior of the factors in Q n (λ, x) as λ → 0 then shows that any singularity of Q n (λ, x) at the origin λ = 0 is removable, and the asymptotic behavior of the same factors in the limit λ → ∞ shows that Q n (λ, x) → I as λ → ∞.

Q n (λ, x) is therefore characterized by its jump condition across the unit circle |λ| = 1. Taking counterclockwise orientation for the circle, the jump condition for Q n (λ, x) reads

Using Proposition 7.4, the jump matrix can be written as

We summarize by writing the Riemann-Hilbert problem for Q n (λ, x).

Riemann-Hilbert Problem 7.5. Fix generic monodromy parameters (e 1 , e 2 ), n ∈ Z, and

with the following properties:

Jump condition: Q n (λ, x) takes analytic boundary values on the unit circle from the interior and exterior, denoted Q n,+ (λ, x) and Q n,-(λ, x) for |λ| = 1 respectively, and they are related by

Henceforth, to avoid the notation becoming unwieldy, we understand that all quantities appearing with subscript n are evaluated at parameter Θ ∞ while quantities appearing with subscript -n are evaluated at parameter Θ 0 .

Having succeeded in removing the problematic jump conditions along rays emanating from 0, ∞ in the λ plane by defining Q n (λ, x), we would next like to consider the limiting behavior of this problem as n → +∞ with x = z/n and z fixed. It is convenient to first renormalize Q n (λ, x), essentially by a transformation that diagonalizes the coefficient µ n B n (x)σ 3 B n (x) -1 of λ -1 in the matrix of the Lax equation (7.5). In other words, in the jump condition for Q n (λ, x) we prefer to replace Φ(∞) n (λ, x) with a suitable left-diagonal multiple of B n (x) -1 Φ(∞) n (λ, x). Observe that the coefficient B n (x) is determined up to right-multiplication by a diagonal matrix by (7.6), in which the second row of the matrix on the right-hand side is

, where we used (7.12) and (7.21). Indeed, the first column b

(1)

n (x) = 0 while the second column b

(2)

n (x) = 0. By selecting specific constant factors for each column, we obtain a matrix P n (x) differing from B n (x) by right-multiplication by a diagonal matrix, and given explicitly by

in which the dependence on the index n enters via (7.12) and (7.27). Then to get the desired modification of the jump matrix we set

Normalization: R n (λ, x) → I as λ → ∞.

The matrices Ξ

n (x) and ∆

n (x) defined in (6.5) and (6.4), respectively, can be expressed in terms of R n (λ, x) as follows:

Here, A n (x) is the matrix coefficient defined in (7.1). We now show that the jump matrix in (7.42) has explicit limits as n → +∞ along even or odd subsequences, with the convergence being uniform for |λ| = 1 and bounded z where x = z/n. To this end, we compute the asymptotic behavior of

) in this setting is (7.33). When |λ| = 1,

Using (7.24), we see that

Now, for x > 0, the principal branch power (-ixλ) σ 3 /2 has the same domain of analyticity as λ σ 3 /2 , and these two analytic functions are related by the identity

Therefore,

Now using (7.31), we have

The diagonal elements in (7.44) can be simplified using the identity (see [38, equation (13.15.3)])

for the (1, 1) entry and µ → -1 2µ n for the (2, 2) entry, and the off-diagonal elements can be simplified using the identity (see [38, equation (13.15.4)

2κ n and µ → 1 2µ n for the (1, 2) entry and µ → 1 2 + µ n for the (2, 1) entry. The result is that

We will need the following result for the large n limit of Whittaker functions appearing here, cf. [38, equation (13.21.1)]. Then the asymptotic formula

holds uniformly in the limit κ → ∞ in any (possibly complex) direction under the assumption ζ = O(1).

Proof . We start from the formula [38, equation (13.14.6)] which holds under the indicated condition on µ:

Clearly, e -ζ/(2κ) = 1-ζ/(2κ)+O κ -2 as κ → ∞ for ζ = O(1), and the product in the summand has the expansion

uniformly for all indices s. This follows from the Fredholm expansion formula Indeed, with r j = -κ -1 (µ + 1 2 + j), we have

and R s ≤ |κ| -1 µ + 1 2 + (s -1) . Therefore, R k s ≤ |κ| µ + 1 2 + (s -1) k holds for all k ≥ 2 whenever |κ| ≥ 1. Consequently,

(7.48)

We apply Lemma 7.7 to (7.45) by taking ζ = -1 2κ n Z = -1 2κ n ixλ. If x = z/n and z = O(1), then using (7.12), we see that ζ = -1 2 izλ + O n -1 holds for |λ| = 1. So, (7.45) becomes the statement that

(principal branch) (7.49) holds in the limit n → ∞ with x = n -1 z uniformly for z = O(1) and |λ| = 1. Similarly, to study the parametrix near 0, it will be convenient to rewrite formula (7.48) in terms of modified Bessel functions:

Replacing n with -n, Θ ∞ with Θ 0 , and λ with -λ -1 and recalling that µ -n = µ n , gives in the same limit,

The jump matrix in (7.42) therefore reads

. (7.51) Expanding (7.41) for large n > 0 gives x κ -n -κn n -n+ 1 2 (-Θ 0 +Θ∞) e n 2 n+ 1 2 (Θ 0 -Θ∞)

Then, by definition, we have

Furthermore, using identities (7.28), (7.34), and Stirling's formula yields

and similarly,

Therefore, the central factor on the right-hand side of (7.51) satisfies

The leading term is independent of n (mod 2) and has unit determinant. This proves the following.

Proposition 7.8. Define the constant matrix which depends only on the even/odd parity of n via µ n , e 1 , e 2 0 , and e 2 ∞ :

Then the following asymptotic formula holds uniformly for |λ| = 1 and z

as n → ∞ along even/odd subsequences. Proposition 7.8 suggests defining the following limiting Riemann-Hilbert problem.

Riemann-Hilbert Problem 7.9 (limiting problem, even/odd subsequences of n).

(7.54)

Note that the Bessel functions J ν (ρ ∞ ) and I ν (ρ 0 ) appearing in the jump matrix in (7.54) are analytic on the unit circle |λ| = 1 except at the point λ = λ c := -ie -i Arg(z) . However, from the identities

and the fact that the indices ν in each column of the Bessel matrix factors in (7.54) differ by 2, combined with the fact that V even/odd is an off-diagonal matrix, one sees easily that

is continuous at λ = λ c and hence is an analytic function of λ on the unit circle. The scalar factor ρ ∞ /ρ 0 is also analytic for |λ| = 1, and therefore the jump matrix in (7.54) is an analytic function of λ when |λ| = 1. At this stage, the existence of a matrix function Reven/odd (λ, z) satisfying Riemann-Hilbert Problem 7.9 is not clear. However, it turns out that there exists a discrete set Σ even/odd ⊂ C such that for z ∈ C \ Σ even/odd , such a matrix does exist and is in fact a meromorphic function of z, see Section 9.1 below. Proof . Noting that Reven/odd (λ, z) necessarily has unit determinant, we form the matrix quotient

Clearly, E n (λ, z) is analytic as a function of λ in the domain of definition, and for each fixed n it tends to I as λ → ∞ as this is true for both R n λ, n -1 z and Reven/odd (λ, z). Across the unit circle, the boundary values of E n (λ, z) are related by

Thus, the jump matrix for E n (λ, z) is the conjugation, by a unit-determinant matrix function of λ independent of n, of the matrix ratio of the jump matrices for R n λ, n -1 z and for Reven/odd (λ, z). But by Proposition 7.8, the latter ratio is I + O n -1 uniformly on the unit circle as n → ∞ along even or odd subsequences. The conjugating factors exist and are uniformly bounded for z in compact subsets of C \ Σ(y 1 , y 2 , y 3 ). It follows that in this limit, Combining (6.6) with (7.43) then shows (7.55). Partly, this works because the dominant term in Ξ

n,12 n -1 z; m is A n,12 n -1 z . ■

In this section we transform Riemann-Hilbert Problem 7.9 to match the form of Riemann-Hilbert Problem 5.1. To this end, using [38, equations (10.4.4) and (10.4.6)] to express the Bessel function J ν (⋄) in terms of the Hankel functions H

(1)

ν (⋄) and the relations [38, equation (10.4.4) and (10.4.6)], H

-ν (⋄) = e πiν H (1) ν (⋄), H

-ν (⋄) = e -πiν H (2) ν (⋄), we arrive at the identity:

To obtain appropriate asymptotic formulae for the matrix on the right-hand side of (7.56), we first apply the identity [38, equation (10.6.1)]

which gives

(7.57)

The matrix on the right-hand side is amenable to asymptotic analysis as ρ ∞ → ∞; using the asymptotics of Hankel functions [38, equations (10.17.5) and (10.17.6)] and (7.57) yields

We turn to analogously treating the final factor of the jump of Reven/odd (λ, z); using [38, equation (10.27.7)] and the above relations, we have

2µn-1 e -πi/2 ρ 0 H

1-2µn e -πi/2 ρ 0 H

(1) 1+2µn e -πi/2 ρ 0 H

(2) -1-2µn e -πi/2 ρ 0   e πiµn e πiµn -e 3πiµn -e -πiµn . This allows us to find the following large-ρ 0 asymptotics:

2µn-1 e -πi/2 ρ 0 H

1-2µn e -πi/2 ρ 0 H

For convenience, we introduce the notation

2µn-1 (⋄) H

(2)

with a fixed determination of the square root; this choice of prefactor guarantees that we have det(H n ) = 1 identically. Using the identity [38, equation (10.11.4)], we note that H n satisfies

We can now rewrite the jump condition (7.54) as

Reven/odd

Next, define

H n e -πi/2 ρ 0 , |λ| < 1. where the jump depends only on the parity of n. Furthermore, since ρ ∞ and ρ 0 change signs across the negative imaginary axis, we may use (7.61) to find

for λ on the negative imaginary axis with |λ| > 1, oriented towards the origin and

for λ on the negative imaginary axis with |λ| < 1, oriented away from the origin. It follows from Riemann-Hilbert Problem 7.9, (7.62), and (7.58) that Ω even/odd has the following asymptotic behavior as λ → ∞:

where the O λ -2 represents an asymptotic series that is differentiable term-by-term with respect to both λ and z. Analogously, we have

where O λ 2 represents an asymptotic series at the origin λ = 0 which is similarly term-by-term differentiable. Notice that we can now relate the limiting formula from Lemma 7.10 to Ω even/odd using definitions (7.62) and (7.60) to find that Reven/odd

-1-2µn e -πi/2 ρ 0 + H

1-2µn e -πi/2 ρ 0 Ω even/odd 21

2µn-1 e -πi/2 ρ 0 + H

2µn+1 e -πi/2 ρ 0 Ω even/odd 22 (λ, z) λ=0 .

Then, using (7.67) and (7.59) yields lim n→∞ n even/odd

To extract the monodromy parameters of U (z) from Ω(λ, z), we notice that it solves Riemann-Hilbert Problem 5.1 with

and 8 Small x asymptotics and proof of Proposition 1.5

Inspired by [33], see also [22,Theorem 3.2], the goal of this section is to compute the asymptotics as x → 0 of the Bäcklund iterates u n (x) for fixed n and, by evaluating at n = 0, arriving at the asymptotic behavior of a generic solution of PIII(D 6 ) in this limit. Observe that the matrices Ξ

n (x) and ∆

n (x) defined in (6.5) and (6.4), respectively, can be expressed in terms of Q n (λ, x) as follows:

Using (6.6) then gives

Also, combining (7.18) and (7.21), we have

in which all x-dependence is explicit. To analyze u n (x) for small x it therefore remains to obtain asymptotics of Q n (λ, x) as x → 0. To do this, let V Q denote the jump matrix (7.40). Then, denoting the off-diagonal constant matrix

we arrive at

where matrices D n , M are defined in (7.24) and (7.31), respectively. To consider the limit x → 0, we start with the Whittaker matrix M(ixλ; κ n , µ n ). Using [38, equation (13.14.6)], we obtain

Therefore, using the definition (7.31) of M(Z; κ, µ) we have

in the limit x → 0 uniformly for |λ| = 1. Similarly, in the same limit,

For x > 0, the functions λ p , -λ -1 p , (-ixλ) p , and ixλ -1 p (the latter two being principal branches) all have the same branch cut, namely iR -. One has the following identities:

(-ixλ) p = e -iπp/2 x p λ p , ixλ -1 p = e iπp/2 x p λ -p , λ -1 p = e iπp λ -p .

It follows that

Because O n is off-diagonal, the central factors in V Q (λ, x) simplify as follows:

Consequently, we have

The matrix V Q (λ, x) does not possess a finite limit as x → 0 due to the factors x (κn-1 2 )σ 3 , x ( 1 2 -κ -n )σ 3 ; this can be handled by introducing the following transformation. Let ς := λx and

It follows that Q n is analytic as a function of ς for ς ∈ C \ {|ς| = 1} and satisfies

Furthermore, on the circle |ς| = 1, the jump condition

where the jump contour has counterclockwise orientation, and

Therefore, Q n (ς, x) solves the following Riemann-Hilbert problem.

Riemann-Hilbert Problem 8.1. Fix generic monodromy parameters (e 1 , e 2 ), and x ∈ C. Seek a 2 × 2 matrix function ς → Q n (ς, x) with the following properties:

Analyticity: Q n (ς, x) is an analytic function of ς for |ς| ̸ = 1.

Jump condition: Q n (ς, x) takes analytic boundary values on the unit circle from the interior and exterior, denoted Q n,+ (ς, x) and Q n,-(ς, x) for |ς| = 1 respectively, and they are related by

The jump V Q has a limit as x → 0, uniformly for |λ| = 1 for |Re µ n | < 1 2 , and satisfies the estimate

By the standard theory of small-norm Riemann-Hilbert problems, we arrive at , Re µ n < 0, (8.8) where ϵ n = sgn(Re µ n ). The concerned reader may note that the leading coefficient in (8.8) is finite due to the genericity conditions on (e 1 , e 2 ) (see the beginning of Section 7). Indeed, assumption (i) guarantees that 2µ n ̸ ∈ Z, condition (ii) requires e 1 e 2 ̸ = 0, and condition (iii) guarantees that

Evaluating the above at n = 0 yields (1.24) and finishes the proof of Proposition 1.5. One notable application of this is to the family of rational solutions of Painlevé-III already discussed at the end of Section 4.2. This corresponds to the choice m = Θ 0 = Θ ∞ -1 and µ 0 = 1/4. It follows from (8.8) that u n (x; m) has a well-defined value at x = 0 which is given by (2.20), (2.21) in the case where n is even or odd, respectively. We can verify that these values are consistent with (8.8) by noting that e 1 , e 2 , e 2 0 , e 2 ∞ are invariant under an even increment n → n + 2, and so we have the general formulae 9 Alternative Riemann-Hilbert problem for Painlevé-III(D 8 )

9.1 Fabry-type transformation and existence of R even/odd (λ, z)

The Lax pair (5.25) is unusual in that its coefficient matrices have non-diagonalizable leading terms at both of its singular points λ = 0 and λ = ∞, i.e., the coefficients of λ 0 and λ -2 in (5.24) are not diagonalizable. To deduce the existence of the matrix functions Ω even/odd (λ, z) and R even/odd (λ, z), we identify this Lax pair with ones appearing in the literature by considering the following Fabry-type transformation

2 , z (-iσ 2 ), π 2 < Arg(ξ) < π, Ω ξ 2 e 5πi 2 , z iσ 2 , -π < Arg(ξ) < -π 2 , (9.1) 9 The case Re(µn) = 0 can be treated similarly, and produces a leading term that is a combination of both leading terms, which we omit for brevity. when |ξ| > 1, and This result also uses the identity (9.4). Using this symmetry, it can be checked that the Fabry transformation (9.1)-(9.2) is invertible with

where all roots are principal branches. While the singular behavior of S(ξ, z) at ξ = 0 is concerning, the fact that the leading coefficient is a singular matrix allows us to handle this problem by letting

where one can carry out a computation similar to the one in Section 5.3 to arrive at a pair of differential equations analogous to (5.25), but with diagonalizable leading matrices at the two singular points at ξ = 0, ∞; this system appears in [33, Chapter 2] and [14], for example. Since we do not make use of this Lax pair, we omit the calculation.

Using (9.4), it follows that which implies that matrix S(ξ, z) also satisfies the symmetry (9.5). To simplify this symmetry, let S(ξ, z) := e -πiσ 3 /4 S(ξ, z)e πiσ S(ξ, z) = I + Ξ (8) (z)ξ -1 + O ξ -2 e i(2z) 1/2 ξσ 3 as ξ → ∞, (9.11) and S(ξ, z) = ∆ (8) (z) I + Π(z)ξ + O ξ 2 e (2z) 1/2 ξ -1 σ 3 as ξ → 0, (9.12)

where ∆ (8) (z) may be written in terms of entries of ∆ (8) (z) and K.

Now, the matrix S(ξ, z) satisfies the symmetry σ 1 S(-ξ, z)σ 1 = S(ξ, z).

Furthermore, it was shown in [33,Theorem 4] that matrix S(ξ, z) exists for all z outside of a discrete set Σ and is a meromorphic function of z in C \ Σ. Since the transformations used to arrive to S(ξ, z) from R(λ, z) and Ω(λ, z) are invertible, we deduce the existence of matrix functions satisfying Riemann-Hilbert Problems 7.9 and 5.1. It was shown in [39] that Σ coincides with the set of zeros of the τ -function associated to the Riemann-Hilbert problem. According to [23] where

and

Similarly, the coefficients in (9.11)-(9.12) have the following expressions ∆ (8) (z) = e -iπ/4 U (z) 1/2 σ 1 , Ξ (8) (z) = i z 2

In our computation we expressed W (z), X(z), and V (z) in terms of U (z) and U ′ (z) using the identities (5. To complete the proof of Theorem 1.4, we must show that U odd (z) = -1/U even (z). One can already observe that this should be the case by checking that the leading behavior predicted in Theorem 1.7 satisfies the involution, but we now present a proof on the level of Riemann-Hilbert problems. First, note that if one chooses the square root in (7.53) in such a way that V even = iσ 3 V odd , it follows from (7.69) that 10

C odd 0∞ = σ 3 C even 0∞ σ 3 .

This, in particular, implies the symmetry S odd (λ, z) = σ 3 S even (λ, z)σ 3 , and, in view of (9.6), we have -iλ T even (z)

2U even (z)X even (z) -4i(U even (z)X even (z)) 2 -i even (z)X even (z) + ρ ∞ (λ, z) 0 i 0 0 . (9.17)

To deduce the relationship between U even , U odd , we now recall that Ω even/odd satisfy the Lax pair (5.25). Transforming Ω even as in the right-hand side of (9.16) induces a gauge transformation of the λ-equation and we have that Ω odd satisfies two equations; the first is the one in (5.25) and the second is

where

Using (9.17) and (5.3), we see that Λ(λ, z) = 0 iz 0 0 + 1 4λ 2 -V even (z) + 8iU even (z)X even (z) F even (z) 2 -2 + V even (z) -8iU even (z)X even (z)

(U even (z)) 2 X even (z) 2i(U even (z)) 3 (X even (z)) 2 iU even (z)/2 -(U even (z)) 2 X even (z) , 10 One can check that making the other choice of the square root yields the same connection matrix but with the opposite sign, and so it follows from Remark 5.2 that this choice is immaterial.