Rational Solutions of the Fifth Painlev´e Equation. Generalised Laguerre Polynomials

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Introduction
The fifth Painlevé equation is given by with α, β, γ and δ constants.In the generic case of (1.1) when δ ̸ = 0, then we set δ = − 1 2 , without loss of generality (by rescaling z if necessary) and obtain which we will refer to as P V .
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 transcendents, i.e. the solutions of the Painlevé equations, can be thought of as nonlinear analogues of the classical special functions.Iwasaki, Kimura, Shimomura and Yoshida [34] characterize the six Painlevé equations as "the most important nonlinear ordinary differential equations" and state that "many specialists believe that during the twenty-first century the Painlevé functions will become new members of the community of special functions".Subsequently the Painlevé transcendents are a chapter in the NIST Digital Library of Mathematical Functions [62, §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 all the Painlevé equations, except P I , possess rational solutions, algebraic 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.[14,23,29] and the references therein.These hierarchies are usually generated from "seed solutions" using the associated B äcklund transformations and frequently can be expressed in the form of determinants.
Vorob'ev [72] and Yablonskii [76] expressed the rational solutions of P II in terms of special polynomials, now known as the Yablonskii-Vorob'ev polynomials, which were defined through a second-order, bilinear differential-difference equation.Subsequently Kajiwara and Ohta [37] derived a determinantal representation of the polynomials, see also [35,36].Okamoto [57] obtained special polynomials, analogous to the Yablonskii-Vorob'ev polynomials, which are associated with some of the rational solutions of P IV .Noumi and Yamada [54] generalized Okamoto's results and expressed all rational solutions of P IV in terms of special polynomials, now known as the generalized Hermite polynomials H m,n (z) and generalized Okamoto polynomials Q m,n (z), both of which are determinants of sequences of Hermite polynomials; see also [38].
Umemura [69] derived special polynomials associated with certain rational and algebraic solutions of P III and P V , which are determinants of sequences of associated Laguerre polynomials.(The original manuscript was written by Umemura in 1996 for the proceedings of the conference "Theory of nonlinear special functions: the Painlevé transcendents" in Montreal, which were not published; see [61].)Subsequently there have been further studies of rational and algebraic solutions of P V [13,17,41,47,52,58,73].Several of these papers are concerned with the combinatorial structure and determinant representation of the generalised Laguerre 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 combinatorial properties [67,68,69].See also [50] and results on the combinatorics of the coefficients of Wronskian Hermite polynomials [8] and Wronskian Appell polynomials [7].
We define generalised Laguerre polynomials as Wronskians of a sequence of associated Laguerre polynomials specified in terms of a partition of an integer.We give a short introduction to the combinatorial concepts in §2 and record several equivalent definitions of a generalised Laguerre polynomial in §3, where we also show that the polynomials satisfy various differential-difference equations and discrete equations.In §4 we express a family of rational solution of P V (1.2) in terms of the generalised Laguerre polynomials.For certain values of the parameter, we show that the solutions are not unique.Rational solutions of the P V σ-equation, the second-order, second-degree differential equation associated with the Hamiltonian representation of P V , are considered in §5, which includes a discussion of some applications.In §6 we describe rational solutions of the symmetric P V system.Properties of generalised Laguerre polynomials are established in §7 as well as an explicit description of all partitions with 2-core of size k and 2-quotient (λ, ∅) for all partitions λ.Then in §8 we obtain the discriminants of the polynomials, describe the patterns of roots as a function of the parameter and explain how the roots move as the parameter varies.Finally, we show that many of the results in the last section can be expressed in terms of combinatorial properties of the underlying partition.We also obtain explicit expressions for the coefficients of Wronskian Laguerre polynomials that depend on a single partition using the hooks of the partition.
A hook length h jk is assigned to box (j, k) in the Young diagram via 3) The hook length counts the number of boxes to the right of and below box (j, k) plus one.Thus , where H λ is the set of all hook lengths.The entries of the degree vector h λ are the hooks in the first column of the Young diagram.Examples of Young diagrams and the corresponding hook lengths are given in Figure 2.1.
A partition can be represented as p + 1 smaller partitions known as the p-core λ and p-quotient (ν 1 , . . ., ν p ).A partition is a p-core partition if it contains no hook lengths of size p.Therefore the example partition (2, 1) is a 2-core and λ = (4 2 , 2, 1 3 ) is both a 6-and 7-core.We only consider p = 2 here.The hooks of size 2 are vertical or horizontal dominoes.We note that all 2-cores are staircase partitions λ = (k, k − 1, . . ., 1).
The 2-core of a partition is found by sequentially removing all hooks of size 2 from the Young diagram such that at each step the diagram represents a partition.The terminating Young diagram defines the 2-core, which we denote λ.It does not depend on the order in which the hooks are removed.For example, the partition (4 2 , 2, 1 3 ) has 2-core λ = (2, 1). Figure 2.1(a) shows that there are three choices of domino that may be removed at the first step.The 2-height ht(λ) (or 2-sign) of partition λ is the (unique) number of vertical dominoes removed from λ to obtain its 2-core.Equivalently, the 2-height is the number of vertical dominoes in any domino tiling of the Young diagram of λ.
The 2-quotient records how the dominoes are removed from a partition to obtain its core.James' p-abacus [31] is a useful tool to determine the quotient, and provides an alternative visual representation of a partition.A 2-abacus consists of left and right vertical runners with bead positions labelled 0, 2, 4, . . .(left) and 1, 3, 5, . . .(right) from top to bottom.To represent a partition on the 2-abacus, place a bead at the points corresponding to each element of the degree vector h.Since a partition can have as many 0's as we like, we allow an abacus to have any number of initial beads and any number of empty beads after the last bead.There are, therefore, an infinite set of abaci associated to each partition, according to the location of the first unoccupied slot.We return to this point below.The parts of a partition are read from its abacus by counting the number of empty spaces before each bead.
A bead with no bead directly above it on the same runner corresponds to a hook of length 2 in the Young diagram.The 2-core λ is found from the abacus by sliding all beads vertically up as far as possible and reading off the resulting partition.Figure 2.1 shows the Young diagram and hooklengths of (4 2 , 2, 1 3 ) in (a), an abacus representation in (c), its 2-core λ = (2, 1) in (b) and the abacus corresponding to λ that is obtained from (c) by pushing up all beads.9 5 3 2 The 2-quotient is an ordered pair of partitions (ν 1 , ν 2 ) that encodes how many places the beads on each runner are moved to obtain the 2-core.The 2-quotient ordering is specified by ensuring the 2-core has at least as many beads on the second runner as the first.One can always add a bead to the left runner of the partition abacus and shift all subsequent beads one place if this condition is not met [74], swapping the order of the quotient partitions.Consequently, the relationship between a partition and its 2-core of size k and 2-quotient (ν 1 , ν 2 ) is bijective.In the running example, one bead on the left runner is moved one place and another bead is moved three places.This is recorded in the partition ν 1 = (3, 1).Only one bead is moved on runner 2, by one space, and so ν 2 = (1).Therefore the 2-core and 2-quotient of λ = (4 2 , 2, 1 3 ) are (2, 1) and ((3, 1), (1)) respectively.
3 Generalised Laguerre polynomials m,n (z), which is a polynomial of degree (m+1)n, is defined by where n (z) is the associated Laguerre polynomial m,n (z) can also be written as the Wronskian m+n (z) .

Using the result
[62, equation (18.9.13)], it can be shown using induction that Hence setting α = µ + n gives and so we obtain Since we can add a multiple of any column to any other column without changing the Wronskian determinant, we keep the last term in each sum: On interchanging the j th column with the (n − j + 1) th column, we find m+n (z) .
Definition 3.4.The elementary Schur polynomials p j (t), for j ∈ Z, in terms of the variables t = (t 1 , t 2 , . ..), are defined by the generating function with p 0 (t) = 1.The Schur polynomial S λ (t) for the partition λ is given by The generalised Laguerre polynomial T where λ = ((m + 1) n ) and with L (α) n (z) the associated Laguerre polynomial.
Remark 3.7.We note that Proof.Apply the standard relation with λ * = (n m+1 ) to the Schur form of the generalised Laguerre polynomial (3.5).
Lemma 3.9.The generalised Laguerre polynomial T (µ) m,n (z) can also be written as the determinants where Proof.These identities are easily proved using the well-known formulae (3.4) and (3.5), and properties of Wronskians in either (3.1) or (3.3).
Lemma 3.10.The generalised Laguerre polynomial T (µ) m,n (z) satisfies the second-order, differentialdifference equation Proof.According to Sylvester [65], see also [48], if A n (φ) is the double Wronskian given by which is now known as the Toda equation.From (3.1) m,n , then we need to show that By definition which proves the result.
(i) Lemma 3.10 can also be proved using the well-known Jacobi Identity [19], sometimes known as the Lewis Carroll formula, for the determinant where D i j is the determinant with the i th row and the j th column removed from D. If and so (3.22) follows from the Jacobi Identity (3.24) with i = k = n and j = ℓ = n + 1.
(ii) We note that the generalised Hermite polynomial with H k (z) the Hermite polynomial, which arises in the description of rational solutions of P IV , satisfies two second-order, differential-difference equations, see [54, equation (4.19)].
The generalised Laguerre polynomial T (µ) m,n (z) satisfies a number of discrete equations.In the following Lemma we prove two of these using Jacobi's Identity (3.24).Lemma 3.12.The generalised Laguerre polynomial T (µ) m,n (z) satisfies the equations Proof.As the n + 1-dimensional determinant in (3.25) and (3.26) is the same, then to apply Jacobi's Identity (3.24), it'll be necessary to use two different representations of m,n+1 .To prove (3.25), we use T (µ) m,n as defined by (3.1) and so we consider Then using Jacobi's Identity (3.24) with i = k = 1 and j = ℓ = n + 1, we obtain (3.25) as required.
To prove (3.26), we use the representation of T (µ) m,n given by (3.3), so we consider and so using Jacobi's Identity with i = k = 1 and j = ℓ = n + 1 gives (3.26) as required.
The generalised Laguerre polynomial T where D z is the Hirota bilinear operator and the discrete bilinear equation Proof.In [70, Theorem 3.6], Vein and Dale prove three variants of the Jacobi Identity (3.24).To prove some to the results in this Lemma, we use, which is identity (C) in [70, Theorem 3.6] with r = 1.For (3.27a), consider the determinants where W n (φ) is defined by and so m,n−1 , which proves the result.
Proof.See Kitaev, Law and McLeod [41]; also [29,Theorem 40.3 Rational solutions in cases (ii) and (iii) of Theorem 4.1 are expressed in terms of generalised Umemura polynomials.As mentioned above, Umemura [69] defined some polynomials through a differential-difference equation to describe rational solutions of P V (1.2); see also [13,52,75].Subsequently these were generalised by Masuda, Ohta and Kajiwara [47], who defined the generalised Umemura polynomial U (α) m,n (z) through a coupled differential-difference equations and also gave a representation as a determinant.Our study of the generalised Umemura polynomials is currently under investigation and we do not pursue this further here.
Rational solutions in case (i) of Theorem 4.1 are special cases of the solutions of P V (1.2) expressible in terms of Kummer functions M (a, b, z) and U (a, b, z), or equivalently the confluent hypergeometric function 1 F 1 (a; c; z).Specifically with n (z) the associated Laguerre polynomial, cf.[62, equation (13.6.19)].Determinantal representations of these rational solutions are given in the following Theorem.
with L (α) n (z) the associated Laguerre polynomial (3.2), then is a rational solution of P V (1.2) for the parameters Proof.This result can be derived from the determinantal representation of the special function solutions of P Proof.From (4.4), by definition . Now we use the identity with Using the recurrence relation cf. [62, equations (18.9.14), (18.9.23)], it is straightforward to show by induction that where b j,k , j = 0, 1, . . ., k − 1 are.)Therefore, using (4.7) and (4.8), we have since, as in the proof of Lemma 3.2, we need only keep the last term due to properties of Wronskians.
is a rational solution of P V (1.2) for the parameters In the case when n = 0 then is a rational solution of P V (1.2) for the parameters Proof.The result follows from Theorem 4.3 and Lemma 4.5.
Corollary 4.7.The rational solutions related through the symmetry S 1 (4.1) are given by with m,n (z) the polynomial given by (3.17), which is a rational solution of P V (1.2) for the parameters In the case when n = 0 then is a rational solution of P V (1.2) for the parameters Proof.Since T It is known that rational solutions of P III can be expressed either in terms of four special polynomials or in terms of the logarithmic derivative of the ratio of two special polynomials [11,Theorem 2.4].Hence it might be expected that the rational solutions of P V discussed here can also be written in terms of the logarithmic derivative of the ratio of two generalised Laguerre polynomials.Remark 4.8.Using computer algebra we have verified for several small values of m and n that alternative forms of the rational solutions (4.10) and (4.12) are given by respectively.Consequently, by comparing the solutions we expect the relations ) where D z is the Hirota bilinear operator (3.28).We envisage that the relations (4.16) can be proved using the Jacobi identity (3.24) or a variant thereof, though we don't pursue this further here.
Setting n = 0 in (4.14) gives , which is (4.11), since The solutions (4.13) and (4.15) in the case when n = 0 can be shown to be the same in a similar way.
Remark 4.9.From Theorem 4.6 we note that w m,n (z; −m − n − j) and w m,j−1 (z; −m − n − j) are both rational solutions for The equality of the solutions follows from lemma 7.2 and the definition of w m,n (z; µ) in the form (4.14).
We add that m w m,n (z;

Non-uniqueness of rational solutions of P V
Kitaev, Law and McLeod [41, Theorem 1.2] state that rational solutions of P V (1.2) are unique when the parameter µ ̸ ∈ Z.In the following Lemma we illustrate that when µ ∈ Z then non-uniqueness of rational solutions of P V (1.2) can occur, that is for certain parameter values there is more than one rational function.
Lemma 4.10.Consider the rational solutions of P V (1.2) given by . (4.17) If µ ∈ Z and µ ≥ −n then there are two distinct rational solutions of P V (1.2) for the same parameters.
5 Rational solutions of the P V σ-equation

Hamiltonian structure
Each of the Painlevé equations P I -P VI can be written as a (non-autonomous) Hamiltonian system for a suitable Hamiltonian function H J = H J (q, p, z).Further, there is a second-order, second-degree equation, often called the Painlevé σ-equation or Jimbo-Miwa-Okamoto equation, whose solution is expressible in terms of the solution of the associated Painlevé equation [32,56].
There is a simple symmetry for solutions of S V (5.4) given in the following Lemma.

Classification of rational solutions of S V
There are two classes of rational solutions of S V (5.4), one expressed in terms of the generalised Laguerre polynomial T (µ) m,n (z), which we discuss in the following theorem, and a second in terms of the generalised Umemura polynomial U (α) m,n (z).
Theorem 5.2.The rational solution of S V (5.4) in terms of the generalised Laguerre polynomial for the parameters ν = (m + 1, −n, m + n + µ + 1). (5.9) Proof.This result can be inferred from the work of Forrester and Witte [25] and Okamoto [58] on special function solutions of S V , together with the relationship between Kummer functions and associated Laguerre polynomials (4.3).We have used Lemma 5.1 as a normalisation.
Corollary 5.3.The rational solution of S V (5.4) in terms of the generalised Laguerre polynomial for the parameters ν = (−m − 1, n, −m − n − µ − 1). (5.11) Remark 5.4.We note that This result follow from the factorisation given in Lemma 7.2 of the T (µ) m,n (z) at certain negative integer values of µ.The third case also follows from the invariance of the Hamiltonian H V (q, p, z) under the interchange of ν 2 and ν 3 .

Non-uniqueness of rational solutions of S V
In §4.2 it was shown that there was non-uniqueness of rational solutions of P V (1.2) in case (i) in terms of the generalised Laguerre polynomial T (µ) m,n (z) when µ is an integer.An analogous situation arises for rational solutions of S V (5.4).(5.13)

Probability density functions associated with the Laguerre unitary ensemble
In their study of probability density functions associated with Laguerre unitary ensemble (LUE), Forrester and Witte [25] were interested in solutions of where M ≥ 0, ℓ ∈ N and µ is a parameter, which is S V (5.4) with parameters ν = (−µ, M, M + ℓ).(5.17) Explicitly, we have (5.19)

Joint moments of the characteristic polynomial of CUE random matrices
In their study of joint moments of the characteristic polynomial of CUE random matrices, Basor et al. [6, equation (3.85)] were interested in solutions of the equation where where B k (z) is the determinant n (z) the associated Laguerre polynomial.Basor et al. [6] remark that equation (5.20a) is degenerate at z = 0, which is a singular point of the equation, and so the Cauchy-Kovalevskaya theorem is not applicable to the initial value problem (5.20).
From (3.21c), we have where the second equality follows from (3.19).In terms of the generalised Laguerre polynomial T (µ) m,n (z), a solution of (5.20) is given by Alternatively, in terms of the polynomial T (µ) m,n (z), a solution of (5.20) is given by For example, suppose that N = 2 and k = 2, then from (5.21) and from (5.24) If we seek a series solution of (5.20) in the form then a 2j are uniquely determined with and a 2j+1 = 0 unless k is an integer.If k is an integer then a 2j+1 = 0 for j < k, a 2k+1 is arbitrary, and a 2j+1 uniquely determined for j > k, as discussed in [6].For example, when N = 2 and k = 2 then 6 Rational solutions of the symmetric P V system From the works of Okamoto [57,58,59,60], it is known that the parameter spaces of P II -P VI all admit the action of an extended affine Weyl group; the group acts as a group of B äcklund transformations.In a series of papers, Noumi and Yamada [49,51,53,55] have implemented this idea to derive a hierarchy of dynamical systems associated to the affine Weyl group of type A N , which are now known as "symmetric forms of the Painlevé equations".The behaviour of each dynamical system varies depending on whether N is even or odd.
The first member of the A 2n hierarchy, i.e.A 2 , usually known as sP IV , is equivalent to P IV and given by ) with constraints The first member of the A (1) 2n+1 hierarchy, i.e.A (1) 3 , usually known as sP V , is equivalent to P V (1.2), as shown below, and given by with the normalisations and κ 1 , κ 2 , κ 3 and κ 4 are constants such that 3) The symmetric systems sP IV (6.1) and sP V (6.2) were found by Adler [1] in the context of periodic chains of B äcklund transformations, see also [71].The symmetric systems sP IV (6.1) and sP V (6.2) have applications in random matrix theory, see, for example, [24,25].
2) gives the system Solving (6.4a) for v, substituting in (6.4b) gives Making the transformation u = 1/(1 − w) in (6.5) yields Analogously solving (6.4b) for u, substituting in (6.4a) gives Then making the transformation v = 1/(1 − w) gives P V (1.2) with parameters As shown above, P V (1.2) has the rational solution in terms of the generalised Laguerre polynomial and so . (6.8) From equations (3.26) in Lemma 3.12 and (3.27c) in Lemma 3.13, with n → n + 1, we have m−1,n+1 , (6.9) with D z the Hirota operator (3.28), and so the solution of equation ( 6.5) is given by In the case when n = 0 then We note that From equation (6.4a), we obtain Depending on the choice of κ 1 and κ 3 , there is a different solution for v. From (6.3), (6.6b) and (6.7b) we obtain Each of these gives a different solution v m,n (z) which we will discuss in turn.
(i) For the parameters κ = (m, −m − n, µ (i) Analogous rational solutions of sP V (6.2) can be derived in terms of the polynomial (ii) Some rational solutions of sP V (6.2) are given in [3,27,28], where a different normalisation of the symmetric system is used.

Non-uniqueness of rational solutions of sP V
As was the case for P V (1.2) and S V (5.4), there is non-uniqueness for some rational solutions of the symmetric system sP V (6.2).We illustrate this with an example.
Hence the associated solutions of sP V (6.2) are and 7 Properties of generalised Laguerre polynomials Remark 7.1.The generalised Laguerre polynomial where which follows from Lemma 1 in [9], and Lemma 7.2.The generalised Laguerre polynomials have multiple roots at the origin when Moreover at such values of µ the polynomials T (µ) m,n (z) factorise as where T using (7.18) with ν = λ = ((m + 1) n ) and α k = n + µ k .We denote by Λ k,m,n the partition that has 2-core k and 2-quotient (λ, ∅).Simplifying the constant term, we obtain (7.19).Moreover (7.23) follows from (7.19) by replacing z with iz and using the well-known relation We determine the degree vector of partition Λ k,m,n from the degree vector ).Put beads in positions 2(m + 1) to 2(m + n) on the left runner and in positions 1 to 2(n + k − 1) + 1 on the right runner.The components of the degree vector of Λ k,m,n correspond to the positions of the beads: Writing the Wronskian Hermite polynomial explicitly in terms of (7.25) gives (7.21),where the Vandermonde determinant in the denominator of the constant (7.22) arises because the components of the degree vector as given in (7.25) are not ordered.The degree vector h Λ k,m,n is obtained by ordering (7.25) from largest value to smallest value.Depending on k, m, n, there are three possibilities corresponding to the three abaci in Figure 7.1.We deduce from the abaci that the degree vector is The description of the partition Λ k,m,n in (7.20) follows from the degree vector using (2.1) with r = 2n + k.
Remark 7.7.In (7.20) we have explicitly described the partition Λ k,m,n with 2-core k and 2-quotient ((m + 1) n , ∅).This result may be of independent interest to those who work in combinatorics.
Remark 7.8.Wronskian Hermite polynomials of the type H Λ K,m,n (z) appear in [27] in their classification of solutions to P V at half-integer values of the associated Laguerre parameter using Maya diagrams.Such diagrams also represent partitions and there is straightforward connection between their results and the ones in this article.The H Λ K,m,n (z) are related to the k = 2 cases studied in §6 of [27]; the k = 3 case therein relates to solutions of generalised Umemura polynomials at half-integer values of the parameter.

Discriminants, root patterns and partitions
In this section we give an expression for the discriminant of the generalised Laguerre polynomials and obtain several results and conjectures concerning the pattern of roots of the generalised Laguerre polynomials in the complex plane.We finish by noting that several of the results can be reframed using partition data.

Roots in the complex plane
In this section we classify the allowed configuration of roots of T ( 1 2 z 2 ) with block E zeros in green, block G in red, block F in orange and block D in blue.We describe how the roots transition between blocks as a function of µ and determine the size of each root block for a given µ when m = 5 and n = 3, before stating the result for all m, n. ) for various µ.We describe the root blocks and transitions between the blocks as µ varies from −16/5 to −61/5.For µ > −4 the roots form two E-type rectangles of size 6 × 3 as shown in the first two images in Figure (8.3).As µ → −4 all roots move towards the imaginary axis.At µ = −4 the innermost column of three zeros from each rectangle have coalesced at the origin and the remaining roots form two rectangles of size 5 × 3. We discuss the detailed behaviour of the coalesecing zeros in the next section.
As µ decreases further, the zeros at the origin emerge as a pair of zeros on the imaginary axis and two complex zeros forming a pair of columns of height two.The coalescing roots move away from as µ decreases, while all other roots return to the origin at each coalesence until they become part of a D-rectangle.The sizes of each root block of T (µ) 5,3 ( 1 2 z 2 ) for µ between each coalescence point is given in Table 8.2.
The family of Wronskian Hermite polynomials with partitions Λ = (m n ) are known as the generalised Hermite polynomials H m,n (z).The roots form m × n rectangles centered on the origin [12,15].
The appearance of rectangular blocks of width m + 1 and height n for large positive and negative k in the root pictures for ) is consistent with Theorem 9.6 and Remark 9.7 of [18].The results therein imply for large k the roots will, up to scaling, be those of a certain Wronskian Hermite polynomial shifted to the right along the real axis, plus the block reflected in the imaginary axis.The numerical investigations in [8] suggest that the relevant Wronskian Hermite polynomial is H m+1,n (z).
m,n ( 1 2 z 2 ) at µ when there are zeros at the origin.

Root coalescences
We now zoom into the origin to investigate precisely how the zeros that coalesce behave as they approach and leave the origin.We start with the example of T    There are two roots on the imaginary axis, two on the real axis and eight in the complex plane, all of which initially move away from the origin.All roots eventually turn around and return to the origin, along with the next set of six zeros from the innermost column of the E-rectangles.We see the petal-like shapes traced out by the complex zeros as µ decreases from −5 to −6.The values of µ at which each set of zeros turn around are different.The remaining plots in Figure 8.6 show the zeros emerging from the origin and those that coalescence for each of the stated µ.Some roots form F-rectangles when µ < −9.
Our numerical investigations reveal that the angles in the complex plane at which the coalescing roots approach the origin and emerge from it can be determined for all m, n, j where µ = −n − j and j = 1, 2, . . ., m + n.Before giving the result for T (µ) m,n (z) as a function of z, we consider an example.
Example 8.6.The roots of T (µ) 2,3 that coalesce at µ = −3 − j − ε for j = 1 . . ., 5 behave as the n th roots of one or minus one as follows: m,n (z) that coalesce at the origin at ε = 0 approach the origin on the rays in the complex plane defined  The roots that coalesce leave the origin on rays that are rotated through Similarly, when n ≤ m the roots coalesce at and emerge from the origin as µ = −n − j ± ε as roots of ±1 according to j k=1 z n+j+1−2k ∓ (−1) n+k , j = 1, 2 . . ., n, (

The role of the partition
In this section we remark that several features of the generalised Laguerre polynomials can be written in terms of partition data, particularly the hooks of the partition λ = (m + 1) n .We first propose an expression for the coefficients of the Wronskian Laguerre polynomials Ω (α) λ (z) for all partitions λ.The result generalises the expression given in Theorem 3 and Proposition 2 in [8] for the coefficients of the Wronskian Hermite polynomials H Λ (z) for the subset of partitions Λ with 2-quotient (λ, ∅).where ht(P) is the number of vertical dominoes in the partition P that has empty 2-core and 2-quotient (ρ, ∅).We remark that Ψ (α) ρ is a polynomial of degree |ρ| in α with leading coefficient (−1) |ρ| .A consequence is that all coefficients of the Wronskian Laguerre polynomial are written through (8.13) in terms of the hooks of partitions.Remark 8.9.We have also generalised Conjecture 8.8 to determinants of Laguerre polynomials of universal character type [42].Such polynomials are defined in terms of two partitions and are generalisations of Wronskian Hermite polynomials H Λ (z) with 2-quotient (λ 1 , λ 2 ).Examples include the generalised Umemura polynomials [47] and the Wronskian Laguerre polynomials arising in [9,21,22,26].A proof of the more general result is under consideration.
We now record some information about the partitions λ = ((m + 1) n ) of the generalised Laguerre polynomial T The multiset can also be written as The constant d The expression for c m,n follows from (8.11) using the degree vector h λ .We now determine Ψ  2,3 encode the behaviour of the roots that coalesce at the origin at µ = −n − j − ε through the polynomials in Conjecture 8.12.When j = 3 the polynomial is (z 5 − 1)(z 3 + 1)(z − 1) and when j = 3 or j = 5 the polynomial is z 3 − 1.

Lemma 3 . 5 .
(z) can be expressed as a Schur polynomial, as shown in the following Lemma.The generalised Laguerre polynomial T (µ) m,n (z) is the Schur polynomial

Lemma 3 . 13 .
(z) satisfies a number of Hirota bilinear equations and discrete bilinear equations.The generalised Laguerre polynomial T (µ) m,n (z) satisfies the Hirota bilinear equations

m+n− 1
= (−1) n−1 d dz T m,n−1 , which proves the result.The other results (3.27c)-(3.27f)are proved in a similar way. 4 Rational solutions of P V 4.1 Classification of rational solutions of P V Rational solutions of P V (1.2) are classified in the following Theorem.

Theorem 4 . 1 .
Equation (1.2) has a rational solution if and only if one of the following holds:

Example 8 . 2 .z 2 )
(z) in the z 2 -plane as a function of µ.Given the symmetry(3.19), the root plot of T (µ) m,n when µ ∈ (−m − n − 1, . . ., ∞) follows from that of T (−µ−2n−2m−2) Figure 8.1 shows the roots of T in the complex plane for various µ.For µ = −35/2 and µ = −6 the non-zero roots form a pair of approximate rectangles of size 5 × 6.When µ = −14 and µ = −8, there are 24 roots at the origin and two rectangles of roots of size 3 × 6.At µ = −17/2 the roots form two rectangles of size 2 × 6 (or possibly 3 × 6), two approximate trapezoids of short base 4 and long base 5 (or 6) centered on the real axis and two triangles of size 2 centred on the imaginary axis.At µ = −25/2 there are four 4-triangles and two 5 × 2 rectangles.Further investigations suggest that the roots of T (µ) m,n ( 1 2 z 2 ) that are away from the origin form blocks in the form of approximate trapezoids and/or triangles near the origin and rectangles further away.We label such blocks E-G as shown in Figure 8.2.We say a rectangle has size d 1 × d 2 if it has width d 1 and height d 2 .A trapezoid of size d 1 × d 2 has long base d 1 and short base d 2 .If d 2 = 1 then we call the resulting (degenerate) trapezoid a triangle.The blocks of roots centered on the real or imaginary axis in approximate rectangles are labelled blocks E and D respectively, and those forming approximate trapezoids are labelled G and F respectively.Figures 8.2b and 8.2c show the zeros of T

Example 8 . 3 .
Figures 8.3 and 8.4  show the roots of T

Figure 8 . 5 :z 2 )
Figure 8.5:The coalescence of the zeros of T (µ) 5,3 ( 1 2 z 2 ) that are closest to the origin shown by overlaying the zero plots as µ tends to µ = −4 (left) and µ = −5 (right).The arrows show the direction in which µ decreases.The solid lines correspond to zeros that arise from the first column of the E-rectangles, and the dashed lines correspond to zeros that arise from the second column of the E-rectangles.

z 2 )
when n ≤ m and j = −n − ⌈µ⌉ ∈ Z. complex zeros that coalesced turn back towards the origin.The lower solid line in the first quadrant shows the movement of the complex root for µ ∈ (4.2105, −4].The upper line shows the root for µ ∈ [−5, 4.2105).At µ ≈ 4.32656, the imaginary zeros also turn back to the origin.The dashed lines show the coalescence of the six zeros in the innermost columns of the E-rectangles for µ from −4 to −5.At µ = −5 all twelve zeros are at the origin.The top right plot in Figure 8.6 shows the twelve zeros as they emerge from the origin as µ decreases from 4.

1 2 πzFigure 8 . 7 :
Figure 8.7:The coalescence of the zeros of T (µ) 2,3 that are closest to the origin shown by overlaying the zero plots as µ approaches µ = −4 (left) and µ = −5 (right) from the right.The black arrows (left) indicate the direction of the root movement as µ → −4 from the right and the red arrows (right) show the roots leaving the origin as µ decreases from −4.The black arrows show the third roots of unity and the red arrows (right) show the third roots of −1.The blue lines in the right figure without arrows correspond to the movement of the roots that approach the origin as µ → −5 − at angles corresponding to the fourth roots of 1 and the square roots of −1.

Figure 8 . 9 :
Figure 8.9:The hooks on the j th diagonal of the Young diagram of T (µ) [41]taev, Law and McLeod [41, Theorem 1.1]give four cases, though their cases (I) and (II) are related by the symmetry (4.2).Kitaev, Law and McLeod[41]also state that µ ̸ ∈ Z in case (iii), but this does not seem necessary, except for uniqueness as discussed in §4.2.Rational solutions in case (i) of Theorem 4.1 are expressed in terms of generalised Laguerre polynomials, which are written in terms of a determinant of Laguerre polynomials and are our main concern in this manuscript.

Table 8 .
2: Size of the root blocks of T

Table 8 . 5
: Conjectured root blocks of T