A constructive proof for the Umemura polynomials of the third Painlev´e equation

We are concerned with the Umemura polynomials associated with rational solutions of the third Painlev´e equation. We extend Taneda’s method, which was developed for the Yablonskii–Vorob’ev polynomials associated with the second Painlev´e equation, to give an algebraic proof that the rational functions generated by the nonlinear recurrence relation which determines the Umemura polynomials are indeed polynomials. Our proof is constructive and gives information about the roots of the Umemura polynomials.


Introduction
The third Painlevé equation (P III ) has the form where ′ = d/dz and α, β, γ and δ are arbitrary parameters.We discuss the Umemura polynomials associated with rational solutions of (1.1) in the generic case when γδ ̸ = 0, so we set γ = 1 and δ = −1, without loss of generality (by rescaling w and z if necessary), and so consider The six Painlevé equations (P I -P VI ), were discovered by Painlevé, Gambier and their colleagues whilst studying second order ordinary differential equations of the form where F is rational in dw/dz and w and analytic in z.The Painlevé equations can be thought of as nonlinear analogues of the classical special functions.Indeed, Iwasaki, Kimura, Shimomura and Yoshida [22] characterize the six Painlevé equations as "the most important nonlinear ordinary differential equations" and state that "many specialists believe that during the twentyfirst century the Painlevé functions will become new members of the community of special functions".Subsequently this has happened as the Painlevé equations are a chapter in the NIST Digital Library of Mathematical Functions [37, Section 32].
The general solutions of the Painlevé equations are transcendental in the sense that they cannot be expressed in terms of known elementary functions and so require the introduction of a new transcendental function to describe their solution.However, it is well known that P II -P VI possess rational solutions and solutions expressed in terms of the classical special functions -Airy, Bessel, parabolic cylinder, Kummer and hypergeometric functions, respectively -for special values of the parameters, see, e.g., [13,17,19] and the references therein.These hierarchies are usually generated from "seed solutions" using the associated Bäcklund transformations and frequently can be expressed in the form of determinants.
Vorob'ev [45] and Yablonskii [46] expressed the rational solutions of P II which arise only when α ∈ Z, in terms of special polynomials, now known as the Yablonskii-Vorob'ev polynomials, that are defined through the recurrence relation (a second-order, bilinear differential-difference equation) with Q 0 (z) = 1 and Q 1 (z) = z.It is clear from the recurrence relation (1.4) that the Q n+1 are rational functions, though it is not obvious that they are polynomials since one is dividing by Q n−1 at every iteration.In fact, it is somewhat remarkable that the Q n are polynomials.Taneda [40], see also [18], used an algebraic method to prove that the functions Q n defined by (1.4) are indeed polynomials.Umemura [42,43] 1 derived special polynomials with certain rational and algebraic solutions of P III , P V and P VI , see also [32,33].Recently there have been further studies of the special polynomials associated with rational and algebraic solutions of P III [1,3,4,11,25,30,31,36,34,44]; a review of rational and algebraic solutions of Painlevé equations is given in [14].Several of these papers are concerned with the combinatorial structure and determinant representation of the polynomials, often related to the Hamiltonian structure and affine Weyl symmetries of the Painlevé equations.Additionally, the coefficients of these special polynomials have some interesting, indeed somewhat mysterious, combinatorial properties [41,42,43].
These special polynomials arise in several applications.For example, the Umemura polynomials associated with rational solutions of P III and P V arise as multivortex solutions of the complex sine-Gordon equation [2,5,6,38], and in MIMO wireless communication systems [10].
We emphasize that the fact that the nonlinear recurrence relation (1.4) generates polynomials also follows from the τ -function theory associated with the theory of Painlevé equations.The τfunctions are in general entire functions.It can be shown that for P II with α = m, the associated τ -function is Consequently, the rational function Q m (z) has to be a polynomial.Taneda [40] and Fukutani, Okamoto and Umemura [18] independently gave a direct algebraic proof, which is one of the first studies of nonlinear recurrence relations for polynomials.In particular, Taneda [40] defined a Hirota-like operator and showed that if f (z) is a polynomial in z, and . This is based on the assumption that each Q m has simple zeros (implying that Q m and Q m−1 have no common zeros), which in turn can be proved using another identity derived from P II , which is proved in [18,40], see also [26].
In this paper, we are concerned with P III (1.2).In this case the recurrence relation is where µ is a complex parameter; see Theorem 2.3 below.The objective is to extend Taneda's method to prove directly and constructively that the rational functions S n (z; µ) defined by (1.5) are indeed polynomials.Note that in (1.5), there is one more term S n dSn dz , and z in the main term implies that the root z = 0 of S n (z; µ), if exists, will accumulate.To employ Taneda's method, we define another Hirota-like operator Also we need one more identity.We find that it is suitable to use the fourth order differential equation satisfied by S n (z; µ) given in [11].This fourth order equation comes from the secondorder, second-degree equation, often called the Painlevé σ-equation, or Jimbo-Miwa-Okamoto equation, satisfied by the Hamiltonian associated with P III given by [23,36] Multiplying (1.6) by 1/z 2 and differentiating with respect to z gives then letting gives [11, p. 9519] This equation is also instrumental in the analysis of the case when z = 0 is a root of S n (z; µ), see Section 4 below.Finally, we remark that this is not the first paper on the direct proof for Umemura polynomials.In 1999, Kajiwara and Masuda [25] were able to express S n (z; µ) in terms of some Hankel determinant of a n × n matrix of polynomials (also known as Schur functions) that can be obtained from an elementary generating function.However, our proof is constructive, giving more information about the order of roots of S n (z; µ).This information was utilized by Bothner, Miller and Sheng [3,4] in their study of the asymptotics of the (scaled) poles and roots of the rational solutions in their so-called "eye-problem".
In Section 2, we describe rational solutions of equation (1.2).In Section 3, we extend Taneda's algebraic proof for equation (1.4) to equation (1.5).In Section 4, we discuss S n (0; µ) since z = 0 is the only location where S n (z; µ) can have a multiple root, and in Section 5, we discuss our results.

Rational solutions of P III
The classification of rational solutions of equation (1.2), which is P III with γ = 1 and δ = −1, are given in the following theorem.Proof .See Gromak, Laine and Shimomura [19, p. 174]; also [30,31].■ Umemura [42,43] derived special polynomials associated with rational solutions of P III (1.2), which are defined in Theorem 2.2, and states that these polynomials are the analogues of the Yablonskii-Vorob'ev polynomials associated with rational solutions of P II [45,46] and the Okamoto polynomials associated with rational solutions of P IV [35].
Theorem 2.2.Suppose that T n (z; µ) satisfies the recurrence relation Proof .See Umemura [42,43]; also [11,25].■ We note that T n (z; µ) are polynomials in ξ = 1/z.It is straightforward to determine a recurrence relation which generates functions S n (z; µ) which are polynomials in z.These are given in the following theorem.
(3) It is trivial to see that each Umemura polynomial S n (z; µ) is monic, and deg (4) The Umemura polynomials S n (z; µ) also arise in the description of algebraic solutions of the special case of P V when γ ̸ = 0 and δ = 0, i.e., see [12,14,15], which is known to be equivalent to P III (1.2), cf.[19,Section 34]. ( which is known as P III ′ (cf.Okamoto [36]) and is frequently used to determine properties of solutions of P III .However, P III ′ (2.2) has algebraic solutions rather than rational solutions [7,30,31].
Kajiwara and Masuda [25] derived representations of rational solutions for P III (1.2) in the form of determinants, which are described in the following theorem.
(1) We note that (2) The relationship between the polynomial S n (z, µ) and the Wronskian τ n (z; µ) is (3) In the special case when µ = 0, then which is straightforward to show by applying induction to (1.5) with µ = 0.
(4) In the special case when µ = 1, then where θ n (z) is the Bessel polynomial, sometimes known as the reverse Bessel polynomial, given by with K ν (z) the modified Bessel function, cf.[8,9,20,27], which arise in the description of point vortex equilibria [39].We note that Bessel functions also arise in the description of special function solutions of P III , see Theorem 2.9.
The recurrence relation (1.5) is nonlinear, so in general, there is no guarantee that the rational function S n+1 (z; µ) thus derived is a polynomial (since one is dividing by S n−1 (z; µ)), as was the case for the recurrence relation (1.4).However, the Painlevé theory guarantees that this is the case through an analysis of the τ -function.A few of these Umemura polynomials S n (z; µ), with µ an arbitrary complex parameter, are given in Table 1.
It is straightforward to determine when the roots of S n (z; µ) coalesce using discriminants of polynomials.
Proof .See Gromak, Laine and Shimomura [19,Section 35]; also [28,44].■ Plots of the roots of the polynomials S n (z; µ) for various µ are given in [11].Initially for µ sufficiently large and negative, the 1  2 n(n + 1) roots of S n (z; µ) form an approximate triangle with n roots on each side.Then as µ increases, the roots in turn coalesce and eventually for µ sufficiently large and positive they form another approximate triangle, similar to the original triangle, though with its orientation reversed.As shown in Theorem 2.10 below, as |µ| → ∞ the roots of S n (z; µ) tend to "triangular structure" of the roots of the Yablonskii-Vorob'ev polynomial Q n (z) which arise in the description of the rational solutions of P II (1.3).
Bothner, Miller and Sheng [3,4] study numerically how the distributions of poles and zeros of the rational solutions of P III (1.3) behave as n increases and how the patterns vary with µ ∈ C (note that they use a different notation to our notation).
It is well known that P II (1.3) arises as the coalescence limit of P III , cf. [21].If in P III (1.2), we let Hence in the limit as ε → 0, (1.2) coalescences to P II (1.3).In the following theorem, it is shown that the Yablonskii-Vorob'ev polynomial Q n (ζ) arises as the coalescence limit of the polynomial S n (z; µ) in an analogous way, see also [14,16].
Theorem 2.10.The Yablonskii-Vorob'ev polynomial Q n (ζ) arises as the coalescence limit of the polynomial S n (z; µ) given by Proof .Since S n (z; µ) satisfies the recurrence relation (1.5), then making the transformation to (1.5) yields the recurrence relation Hence in the limit as ε → 0, then this coalescences to the equation which is the recurrence relation for the Yablonskii-Vorob'ev polynomial Q n (ζ), recall (1.4).Further, since S 0 (z; µ) = 1 and , for all n, as required.■ Remarks 2.11.
(1) It is not obvious that R n (ζ; ε) is a polynomial in ε, as well as a polynomial in ζ.See Lemma A.1 for a proof.We give the first few R n in Table 3. (2) Masuda [29, Section A.2] discusses the coalescence limit of Umemura polynomials to Yablonskii-Vorob'ev polynomials through the associated Hamiltonians.
Using the Hamiltonian formalism for P III , it is shown in [11] that the polynomials S n (z; µ) satisfy a fourth order bilinear equation and a sixth order, hexa-linear (homogeneous of degree six) difference equation.

Application of Taneda's method
In this section, we use the algebraic method due to Taneda [40] to prove that the rational functions S n (z; µ) satisfying (1.5) are indeed polynomials, assuming that all the zeros of S n (z; µ) are simple.
We define an operator L z as follows: Lemma 3.1.Let f (z) and g(z) be arbitrary polynomials.Then so the result is valid.
Proof .We first prove part (b).If S N (z; µ) and S N −1 (z; µ) have the same root z 0 ̸ = 0, then by (1.5), z 0 is also a root of and hence also a root of dS N dz (z; µ).This implies z 0 is (at least) a double root of S N (z; µ), which contradicts our assumption about S N (z; µ).
Let f be S N −1 .Then n = N − 1 and h = S N S N −2 in Lemma 3.1.Then (3.1) becomes Then by (3.1) and (b) with n = N − 1, we have So, according to (1.5), S N +1 is a polynomial by induction.■ 4 Roots of S n (z; µ) In this section we initially discuss S n (0; µ) since z = 0 is the only location where S n (z; µ) can have a multiple root.
Therefore, we have and so (4.1) is also valid.■ Corollary 4.2.
(a) For all n ∈ N, where for 0 ≤ j < k, , where g n is a polynomial of degree n − 1.
Remark 4.3.Part (a) above means that z = 0 is a root of S n (z; µ) if and only if µ = 0, ±1, ±2, . . ., ±(n − 1), i.e., |µ| is an integer strictly less than n.In particular, the first few ϕ n (µ) are Proof .It is trivial to verify by induction hypothesis, with the help of above and (4.1) that, This proves (a).Also we have Hence by (4.2), where g 2k is a polynomial of degree 2k − 1.Similarly, by (4.2) again, where g 2k+1 is a polynomial of degree 2k.Thus the proof of (b) is complete.■ Proof .We shall make use of the identity (4.6) again.Suppose z 0 is a nonzero root of S n (z; µ).Then from (4.6), Hence if z 0 is a root of dS n dz , then it also has to be a root of d 2 S n dz 2 .That is, if z 0 is not a simple root of S n (z; µ), then its order k ≥ 3. Analyzing on the identity (4.6), the term has the zero z 0 with order at least 2k − 4, while the other terms has order at least 2k − 3. Therefore, let S n (z; µ) = (z − z 0 ) k g(z), where g(z) is a polynomial and g(z 0 ) ̸ = 0. Then there exists a polynomial h(z) such that But the expression inside the bracket must have z 0 as a root.This gives a contradiction.We see that every nonzero z 0 is at most a simple root.This proves (a).Part (b) follows directly from Remark 4.3 and Theorem 3.2.■ These results are illustrated in Figure 1, where plots of S n (z; µ) with µ = 10 (blue) and µ = −10 (red), for n = 2, 3, . . ., 10 are given.Similar figures appear in [11].
Since a 1 = µa 0 , we deduce that (1) Thus when n is large, S n (z; µ) has O n 2 roots, counted according to multiplicity.
µ ∈ Z, then most roots are located at z = 0, while there are only O(n) non-zero roots, and all of them are simple roots.This explains the phenomenon that when µ ∈ Z, the roots and poles of the rational solution w n are unusually fewer than the other µ's nearby, as observed in [3,4].
Next we show that g ℓ+1 (z) is a polynomial.From the proof of Theorems 4.4 and 3.2 (b), we know that all nonzero roots of S n (z; µ) and S n−1 (z; µ) are simple and not common.We conclude that g ℓ−1 divides ∆, which implies that g ℓ+1 is indeed a polynomial.It means S n+1 = z σ ℓ+1 g ℓ+1 (z) is indeed a polynomial.Consequently, parts (a) and (b) follow by induction.■

Conclusions
We have given a direct algebraic proof that the nonlinear recurrence relation (1.5) generates polynomials S n (z; µ), rather than rational functions without direct resort to the τ -function theory of Painlevé equations.However we critically needed a higher order equation derived from the corresponding σ-equation, which seems to be inevitable in the nonlinear scenario.We believe that the method can be developed to apply to the fifth Painlevé equation (P V ) as well, though we shall not pursue this further here.
A About the coalescence limit We claim that V n is a polynomial in ε −1 , and deg V n , ε −1 = 1 2 n(n + 1), so that each R n defined above, as a rational function, is indeed a polynomial in ε.
We emphasize that in the above argument, the terms in S n (z, µ) can achieve maximum power of ε −1 only at those terms involving ζ n(n+1)/2−3k , so that for each derivative with respect to ζ, the power of ε −1 will decrease by one.Also the coefficient at the maximum power of ε −1 does not vanish because of the expression ζ n 2 +n+1 in the third term above.
The proof is now complete.■
where the symbol | means that the right-hand side is divisible by the left-hand side.Proof .(a) This follows directly from the definition.(b) We observe that