Extended Symmetry of Higher Painlevé Equations of Even Periodicity and Their Rational Solutions

Abstract

The structure of the extended affine Weyl symmetry group of higher Painlevé equations of N periodicity depends on whether N is even or odd. We find that for even N, the symmetry group ${\widehat{A}}_{N-1}^{\left(1\right)}$ contains the conventional Bäcklund transformations $s_j,j=1,\dots ,N$, the group of automorphisms consisting of cycling permutations but also reflections on a periodic circle of N points, which is a novel feature uncovered in this paper. The presence of reflection automorphisms is connected to the existence of degenerated solutions, and for $N=4$, we explicitly show how even reflection automorphisms cause degeneracy of a class of rational solutions obtained on the orbit of the translation operators of ${\widehat{A}}_3^{\left(1\right)}$. We obtain the closed expressions for the solutions and their degenerated counterparts in terms of the determinants of the Kummer polynomials.

1. Introduction

The Painlevé equations are six non-linear second-order ODEs, first explored by Paul Painlevé and collaborators in the early 20th century, characterized by their common property that their moveable singularities are poles. Today, they play an important role in the theory of integrable systems with many significant applications to various physical and non-linear systems, such as, for example, random matrix theory, the Ising model, conformal field theory and diffusion processes.

The fifth Painlevé equation (denoted by Painlevé V or PV equation) for $y=y\left(x\right)$ is expressed as follows:

$$y_{xx}=-{\displaystyle \frac{y_x}{x}}+\left({\displaystyle \frac{1}{2y}}+{\displaystyle \frac{1}{y-1}}\right)y_x^2+{\displaystyle \frac{{(y-1)}^2}{x^2}}\left(\alpha y+{\displaystyle \frac{\beta }{y}}\right)+{\displaystyle \frac{\gamma }{x}}y+\delta {\displaystyle \frac{y(y+1)}{y-1}},$$

(1)

with $\alpha ,\beta ,\gamma ,\delta$ being constants. The non-zero $\delta$ can always be normalized to $-1/2$.

Rational solutions of the Painlevé equations are best examined within the symmetric version of these equations, which we here introduce by the following two steps: First, we rewrite a single PV equation as two first-order differential equations (see [1] for slightly different implementation of a similar idea):

$$\begin{array}{cc}\hfill F_x& =F(F-1)(2G-1)-{\displaystyle \frac{{\alpha }_1+{\alpha }_3}{2x}}F+{\displaystyle \frac{{\alpha }_1}{2x}},\hfill \\ \hfill G_x& =-G(G-1)(2F-1)+{\displaystyle \frac{{\alpha }_1+{\alpha }_3-2}{2x}}G+{\displaystyle \frac{{\alpha }_2}{2x}}.\hfill \end{array}$$

(2)

Eliminating G from Equation (2), we find that $y=F{(F-1)}^{-1}$ indeed satisfies the Painlevé V Equation (1) with

$$\alpha ={\displaystyle \frac{1}{8}}{\alpha }_3^2,\beta =-{\displaystyle \frac{1}{8}}{\alpha }_1^2,\gamma ={\displaystyle \frac{{\alpha }_4-{\alpha }_2}{2}},\delta =-{\displaystyle \frac{1}{2}}.$$

Next, we rewrite the Equation (2) as the manifestly $A_3^{\left(1\right)}$ invariant system of four first-order non-linear equations

$$\begin{array}{cc}\hfill g_{i,x}& =g_i{g}_{i+2}(g_{i-1}-g_{i+1})-{\displaystyle \frac{{\alpha }_i+{\alpha }_{i+2}}{2x}}{g}_i+{\displaystyle \frac{{\alpha }_i}{2x}}\hfill \\ \hfill & =g_i{g}_{i+2}(g_{i-1}-g_{i+1})+{\displaystyle \frac{{\alpha }_i}{2x}}{g}_{i+2}-{\displaystyle \frac{{\alpha }_{i+2}}{2x}}{g}_i,i=1,2,3,4\hfill \end{array}$$

(3)

in terms of $g_1=F,g_2=G,g_3=1-F,g_4=1-G$ and with ${\alpha }_4=2-{\sum }_{i=1}^3{\alpha }_i$ so that $g_1+g_3=g_2+g_4=1$ and ${\sum }_{i=1}^4{\alpha }_i=2$. Furthermore, in Equation (3), we assumed the periodicity conditions $g_{i+4}=g_i,{\alpha }_{i+4}={\alpha }_i$.

More broadly, we are interested in the $A_{N-1}^{\left(1\right)}$ symmetric Painlevé equations, here, rewritten as,

$$\begin{array}{cc}\hfill g_{i,x}& =g_i\sum _{k=1}^{{\displaystyle \frac{N}{2}}-1}{g}_{i+2k}\left(-\sum _{r=1}^k{g}_{i+2r-1}+\sum _{s=k}^{{\displaystyle \frac{N}{2}}-1}{g}_{i+2s+1}\right)\hfill \\ \hfill & -{\displaystyle \frac{g_i}{2x}}\left(\sum _{k=0}^{{\displaystyle \frac{N}{2}}-1}{\alpha }_{i+2k}\right)+{\displaystyle \frac{{\alpha }_i}{2x}},i=1,\dots ,N,\hfill \end{array}$$

(4)

which generalize to higher ($N>4$) even N of the symmetric Painlevé V Equation (3) with $A_3^{\left(1\right)}$ symmetry.

The variables $g_i$ and ${\alpha }_i$ satisfy the periodicity conditions $g_{i+N}=g_i,{\alpha }_{i+N}={\alpha }_i$, as well as the conditions

$$\sum _{r=1}^{N/2}{g}_{2r}=1,\sum _{r=1}^{N/2}{g}_{2r+1}=1,\sum _i^N{\alpha }_i=2.$$

(5)

In references [2,3], the higher Painlevé equations were introduced as a system of differential equations that is invariant under $A_{N-1}^{\left(1\right)}$-symmetry and were presented in terms of the variable z such that $x=-z^2/2$.

A common feature of the Painlevé equations is that they emerge as similarity reductions of the soliton equations, as shown in references [4,5,6]. Airault [7] extended this work by deriving a hierarchy of higher-order ODEs, starting with the second Painlevé equation, through similarity reductions of the KdV and modified KdV hierarchies. Kudryashov [8] further explored these ideas, deriving both the first and also second Painlevé hierarchies described earlier in [7]. It is relevant for this study that self-similarity derivation also holds for the type of higher Painlevé Equation (4) we consider here. Crucial for this observation is that these equations can be derived from the formalism of the dressing chain Equation (7) with even N periodic conditions. Meanwhile, the periodic dressing chain equations have been identified with a self-similarity limit of the second flow of the $sl\left(N\right)$ mKdV hierarchy [9,10] for all integers N. This, in turn, establishes the $A_{N-1}^{\left(1\right)}$ invariant Painlevé equations as a self-similarity limit of an integrable mKdV flow.

Equation (4) is manifestly invariant under the Weyl group of Bäcklund symmetry transformations $s_i,i=1,2,\dots ,N$ (10) as well as the group of automorphisms $\{1,\pi ,\dots ,{\pi }^{N-1},{\widehat{P}}_1,$$\dots ,{\widehat{P}}_N\}$ that realize the dihedral group $D_N$ (we thank the anonymous referee for pointing out to us the connection between the group of automorphisms we describe and the dihedral group), which is the group of rotations and reflections of the plane which preserve a regular polygon with N vertices. Here, $\pi$ is a cyclic permutation defined in (10) with ${\pi }^i$ corresponding to rotations within the $D_N$ group and ${\widehat{P}}_n$ are the reflection automorphisms generated by the reflections $P_n$ defined in the Relations (11) and (12) and corresponding to the reflections or “flip” operators of the N-gon within the $D_N$ group. These reflections induce actions of ${\widehat{P}}_n$ on $g_i\left(x\right),{\alpha }_i$ and on the Bäcklund symmetry transformations according to the Relations (13) and (19).

The “even” reflections associated with the even points $n=2i$ of the periodic circle when acting on solutions generated from the particular seed solution by translation operators with the period N are shown to produce ambiguity that here refers to the existence of two solutions that share the same parameters ${\alpha }_i$. In physics, it is common to refer to such ambiguity as (two-fold) degeneracy and it is also common to find that such degeneracy in the energy levels of physical systems has its origin in the underlying symmetry, e.g., the quantum transverse Ising model, which exhibits two-fold ground-state degeneracy and has ${\mathbb{Z}}_2$ symmetry. We will adapt such nomenclature here and refer to solutions that correspond to the same parameters ${\alpha }_i$ as degenerated solutions. In previous papers [11,12], we were able to explain the degeneracy encountered among rational solutions of the Painlevé equation as being caused by the divergence arising when applying some translation operators or their inverse on the particular seed solution (22). Here, we provide an alternative explanation in terms of the ${\mathbb{Z}}_2$ symmetry induced by even reflections of the $D_4$ symmetry group.

Let us note that a similar phenomenon was also found in the context of the matrix second Painlevé hierarchy equations [13,14], where it was observed that by acting with auto-Bäcklund transformations on the same seed solution, it is possible under some conditions to obtain different solutions for the same parameter value.

To avoid confusion, it needs to be pointed out that the degeneracy we discuss has no connection to coalescence (the reducing of one of Painlevé equation to the other Painlevé equation in a special limit; see, for example, [1] or [15]), which sometimes has also been referred to as degeneracy.

The paper is organized as follows: In Section 2, we derive the higher ${\widehat{A}}_{N-1}^{\left(1\right)}$ symmetric Painlevé equations from the dressing chain equations of even periodicity and proceed in Section 3 to analyze their symmetries. In addition to the Bäcklund transformations $s_i,i=1,\dots ,N$ of the extended Weyl group ${\widehat{A}}_{N-1}^{\left(1\right)}$, we introduce reflection transformations $P_n,n=1,\dots ,N$ that together with the cyclic permutations $\{1,\pi ,{\pi }^2,\dots ,{\pi }^{N-1}\}$ form the dihedral group $D_N$ of automorphisms of the higher Painlevé equations of even N periodicity. The reflection automorphisms $P_n\left(k\right)=n-k$ modulo N for $n,k$ being points on the $1,\dots ,N$ periodic circle, are called even and odd reflections, depending on whether n is even or odd. All reflection automorphisms square to one and transform the underlying variable x to $-x$. Further, we describe representation of the reflection automorphisms on the $A_{N-1}^{\left(1\right)}$ symmetries $s_i,i=1,\dots ,N$ and on the abelian subgroup of $A_{N-1}^{\left(1\right)}$, consisting of the translation operators $T_i,i=1,\dots ,N$ [3]. These results pave the way for improved understanding of how even reflections introduce degeneracy among a class of rational solutions of the Painlevé equations. This mechanism is explicitly demonstrated for the $N=4$ model.

Working with the $N=4$ case, we present in Section 4 an explicit construction of a class of rational solutions of the Painlevé Equation (3) that reside on the orbit of the translation operators of the extended affine Weyl group $A_3^{\left(1\right)}$. These solutions are given by two components $F_m^k(x,\mathsf{a}),G_m^k(x,\mathsf{a})$ and are labeled by two independent positive integer parameters that enter the formalism as powers of the translation operators. In Section 5, we find explicit determinant expressions for $F_m^k(x,\mathsf{a}),G_m^k(x,\mathsf{a})$ in terms of the Kummer polynomials for all orbit solutions generated from the particular seed solution (22).

Our technique relies on the observation that the actions of translation operators generate recurrence relations. We solve these recurrence relations in terms of the determinants of the Kummer polynomials, constructed explicitly in Section 5. It is important to highlight the role of one of the odd reflection automorphism, which introduces a “duality” relation. This duality ensures the intrinsic consistency of rational solutions by allowing for two dual formulations based on related definitions of the Kummer polynomials.

In Section 6, we discuss the degeneracy that exists for a class of rational solutions of $N=4$ linking it to actions of the two even reflection automorphisms that are shown to map the rational solutions to different solutions that share the values of the parameters for the $N=4$ Painlevé equations.

The two remaining odd reflection automorphisms are discussed in Section 7. The ${\widehat{P}}_1$ automorphism is shown to connect two dual solutions. The invariance under ${\widehat{P}}_1$ ensures the validity of the recursive relations and accordingly proves the intrinsic consistency of construction of the determinant solutions based on these recursive relations.

In Section 8, we show an example that illustrates how our formalism extends to $N=6$, with the presence of a reflection automorphism being a crucial feature when merging single-orbit solutions into a general solution.

A brief summary and future plans are offered in Section 9.

2. Dressing Chain Derivation of Painlevé Equations for Even Periodicity

We start with a formalism of the dressing chain equations of even periodicity. The conventional definition of a dressing chain of N periodicity [16,17] is

$${(j_n+j_{n+1})}_z=-j_n^2+j_{n+1}^2+{\alpha }_n,n=1,\dots ,N,j_{N+i}=j_i,$$

(6)

However, for even N, this expression requires, for consistency, an imposition of a quadratic constraint that can be introduced as a modification of the dressing chain formulation. Such a modification was proposed on the basis of the Dirac reduction method in [18], where the authors put forward a system of dressing chain equations of even periodicity defined as follows:

$${(j_i+j_{i+1})}_z=-j_i^2+j_{i+1}^2+{\alpha }_i+{(-1)}^{i+1}{\displaystyle \frac{(j_i+j_{i+1})}{\mathsf{\Phi }}}\mathsf{\Psi },i=1,2,\dots ,N,j_{N+i}=j_i,$$

(7)

where

$$\mathsf{\Psi }=\sum _{k=1}^N{(-1)}^{k+1}\left(j_k^2-{\displaystyle \frac{1}{2}}{\alpha }_k\right),\mathsf{\Phi }=\sum _{k=1}^N{j}_k=z.$$

(8)

Remarkably, the dressing chain Equation (7) can be rewritten entirely in terms of $f_i=j_i+j_{i+1}$ without any references to $j_i$. It needs to be emphasized that such elimination of $j_i$ variables while expressing the dressing chain equations in terms of $f_i$ requires inserting the definition of $\mathsf{\Psi }$ from (8) into Equation (7). This is because the relation $f_i=j_i+j_{i+1}$ is not invertible for even N. In contrast, when N is taken to be odd, such relation is invertible, as illustrated by the example of $N=3$, with the inverse relation given by:

$$j_1={\displaystyle \frac{1}{2}}(f_1-f_2+f_3),j_2={\displaystyle \frac{1}{2}}(f_2-f_3+f_1),j_3={\displaystyle \frac{1}{2}}(f_3-f_1+f_2),$$

and after inserting these relations back into Equation (6), one obtains the symmetric Painlevé IV equations $f_{i,z}=f_i(f_{i+1}-f_{i-1})+{\alpha }_i,i=1,2,3$.

In the case of Equation (7), introducing new variables

$$g_i\left(x\right)={\displaystyle \frac{f_i}{z}}={\displaystyle \frac{j_i+j_{i+1}}{z}},x=-z^2/2,$$

(9)

we are able to derive the N even Painlevé Equation (4), where $g_i$ and ${\alpha }_i$ satisfy conditions (5).

Equation (4) agrees with the higher Noumi–Yamada Equation [2], with $A_l^{\left(1\right)}$ symmetry for $l=2n+1$, which were originally written in terms of the variable z. One of the advantages of using x is that it makes it possible to uncover a dihedral group symmetry structure of N automorphisms. These automorphisms preserve the $A_{N-1}^{\left(1\right)}$ Weyl symetry group of the Bäcklund transformations of Equation (4) and will explain the presence of denegeneracy among rational solutions. In the case of equations with $N=4$, we obtain a regular Painlevé V equation directly from (4) as the Painlevé V equation is conventionally expressed using the variable x (see Equation (2) for the derivation).

3. Symmetries of Higher Painlevé Equations for Even N

Equation (4) are invariant under the extended affine $A_{N-1}^{\left(1\right)}$ Weyl group of the Bäcklund transformations, $s_i$ and an automorphism $\pi$:

$$\begin{array}{cc}\hfill s_i& :g_{i\pm 1}\to g_{i\pm 1}\mp {\displaystyle \frac{{\gamma }_i}{g_i}},g_i\to g_i,g_j\to g_j,j\ne i,i\pm 1,\hfill \\ \hfill & {\gamma }_i\to -{\gamma }_i,{\gamma }_{i\pm 1}\to {\gamma }_{i\pm 1}+{\gamma }_i,{\gamma }_j\to {\gamma }_j,j\ne i,i\pm 1,i=1,\dots ,N,\hfill \\ \hfill \pi & :\pi \left(g_i\right)=g_{i+1},\pi \left({\gamma }_i\right)={\gamma }_{i+1},\hfill \end{array}$$

(10)

where, for convenience, we introduced the variables:

$${\gamma }_i={\displaystyle \frac{{\alpha }_i}{2x}}$$

with the normalization ${\sum }_i^N{\gamma }_i=1/x$ that corresponds to the normalization ${\sum }_i^N{\alpha }_i=2$ for the parameters ${\alpha }_i$.

In addition to the extended affine Weyl group of the Bäcklund transformations, the Equation (4) are invariant under automorphisms that are generated by reflections associated to the fixed point n on a periodic circle $1,2,3,\dots , N+1, N+2,\dots$ with a periodic condition $k=k+N$ for any point k. These automorphisms also map $x\to -x$.

We denote these automorphisms as $P_n^N$, but will drop the upper script N, when it is obvious from the context and will use only the $P_n$ symbol. We distinguish between two types of automorphisms, depending on whether n is even or odd. Accordingly, the two classes of reflections are as follows:

• Even reflections associated with even points $n=2i$ denoted by $P_{2i},i=0,1,\dots ,{\displaystyle \frac{N}{2}}$ that act as follows:

  $$P_{2i}\left(k\right)=2i-k,k=0,1,2,3,4,5,\dots ,k+N=k$$

  (11)

  on a periodic line of points with a period of N. Each reflection is equivalent to a single transposition that interchanges k with $2i-k$.
  There are always two fixed points, namely, i and $i+{\displaystyle \frac{N}{2}}$ that according to

  $$P_{2i}\left(i\right)=2i-i=i,P_{2i}(i+{\displaystyle \frac{N}{2}})=2i-i-{\displaystyle \frac{N}{2}}=i-{\displaystyle \frac{N}{2}}\sim i+{\displaystyle \frac{N}{2}},$$

  are being transformed into themselves. The symbol ∼ is to indicate that the periodic condition $k=k+N$ was used. The remaining $N-2$ points fall into $N/2-1$ (even/even and odd/odd) pairs that are mapped into each other under even reflections.
• Odd reflections associated with odd points $n=2i+1$ denoted by $P_{2i+1},i=0,1,\dots$, ${\displaystyle \frac{N}{2}}-1$ that act through

  $$P_{2i+1}\left(k\right)=2i+1-k,k=0,1,2,3,4,5,\dots ,k+N=k$$

  (12)

  on a periodic line of points $k=1,2,3,4,\dots$ with a period of N. Each of these reflections is equivalent to transpositions of $N/2$ pairs of even/odd points that are being interchanged into each other.

All these reflections obviously square to one:

$$P_n^2=1.$$

All the above reflections naturally induce transformations of ${\alpha }_k,g_k$ through:

$$\begin{array}{cc}\hfill {\widehat{P}}_n\left(g_k\left(x\right)\right)& =g_{P_n\left(k\right)}(-x)=g_{n-k}(-x),\hfill \\ \hfill {\widehat{P}}_n\left({\alpha }_k\right)& ={\alpha }_{P_n\left(k\right)}={\alpha }_{n-k},n=2i,2i+1\hfill \end{array}$$

(13)

that keep Equation (4) invariant when also $x\to -x$. The straightforward although formal way to prove the invariance of the Painlevé Equation (4) is to introduce the transformations:

$${\widehat{P}}_n\left(j_k\right)=\sqrt{-1}{j}_{n-k+1},{\widehat{P}}_n\left({\alpha }_k\right)={\alpha }_{n-k},{\widehat{P}}_n\left(z\right)=\sqrt{-1}z.$$

(14)

of the dressing Equation (7) that generate transformations (13) together with $x\to -x$. It follows from these definitions that

$${\widehat{P}}_n\left(\mathsf{\Psi }\right)={(-1)}^n\mathsf{\Psi },{\widehat{P}}_n\left(\mathsf{\Phi }\right)=\mathsf{\Phi },$$

when use is made of the periodic conditions. As a result, the dressing chain Equation (7) remain invariant under the transformations (14). This, in turn, proves the invariance of Equation (4) under the automorphisms (13) that are generated by reflections on a periodic circle. Note that the imaginary factor present in definition (14) goes away when the transformation is applied on the quantities x and $g_k\left(x\right)$.

We find from the definitions of reflections the following relations for obtaining the automorphisms $\pi$ and its powers out of products of reflections ${\widehat{P}}_n$:

$$\pi ={\widehat{P}}_{n+1}{\widehat{P}}_n,{\pi }^{-1}={\widehat{P}}_n{\widehat{P}}_{n+1},{\pi }^2={\widehat{P}}_{n+2}{\widehat{P}}_n,{\pi }^{-2}={\widehat{P}}_n{\widehat{P}}_{n+2},n=2i,2i+1.$$

(15)

More generally, we find the following identities:

$${\widehat{P}}_{n_1}{\widehat{P}}_{n_2}={\pi }^{n_1-n_2},{\widehat{P}}_{n_1}{\pi }^{n_2}={\widehat{P}}_{n_1-n_2},$$

(16)

which show that the reflections and $\pi$ automorphisms close under multiplications and form the group with identity $1={\pi }^0={\widehat{P}}_n^2$. This structure is consistent with the reflection automorphisms transforming $x\to -x$, despite the fact that $\pi$ is not acting on x. We will refer to the reflections and $\pi$ and its powers as the dihedral group of automorphisms with the cyclic permutations $\{1,\pi ,{\pi }^2,\dots ,{\pi }^{N-1}\}$ forming a subgroup of these automorphisms.

Even/odd reflections commute among themselves

$$[P_{2i},P_{2j}]=0,[P_{2i+1},P_{2j+1}]=0,i\ne j$$

while we have the following conjugation transformations:

$$\begin{array}{cc}\hfill P_{2i+1}{P}_{2i}{P}_{2i+1}& =P_{2i+2},P_{2i+1}{P}_{2i}{P}_{2i+3}=P_{2i}\hfill \\ \hfill P_{2i}{P}_{2i+1}{P}_{2i}& =P_{2i+1},P_{2i}{P}_{2i+1}{P}_{2i+2}=P_{2i-1}\hfill \end{array}$$

(17)

for the mixed even/odd reflections.

Additional important identities are as follows:

$${\widehat{P}}_n\pi {\widehat{P}}_n={\pi }^{-1},{\widehat{P}}_n{\pi }^{-1}{\widehat{P}}_n=\pi ,n=2i,2i+1.$$

(18)

It is important that the reflections preserve the Bäcklund symmetries through the following conjugation formula:

$${\widehat{P}}_n{s}_m{\widehat{P}}_n=s_{P_n\left(m\right)},n=2i,2i+1,n,m=1,\dots ,N$$

(19)

where $P_n\left(m\right)$ is the point obtained from m by acting with the reflection $P_n$.

The above situation differs from the one which is encountered in the case of odd N. First, in such a case, the corresponding Painlevé equations remain expressed in terms of the variable z that causes a presence of imaginary factors when dealing with the transformation defined in (14). Extending the definitions of reflections given in Relations (11) and (12) to $N=3$ will not induce any symmetry of the symmetric Painlevé IV equations (without introducing a transformation of the variable z involving an imaginary factor or corresponding change of sign of ${\sum }_i{\alpha }_i$). A similar, albeit more general, class of automorphisms has recently been discussed in [15].

4. A Special Case of $\mathit{N}=\mathbf{4}$ and Painlevé V Equations

Let us consider Equation (3) that are a special $N=4$ case of Equation (4). In this setting, the $A_3^{\left(1\right)}$ extended affine Weyl group of the Bäcklund transformations, $\pi ,s_i,i=1,\dots ,4$ emerges as a group of symmetry operations on Equation (2) and can be obtained from the Relation (10) for $N=4$.

Below, we will explicitly derive the solutions of Equation (2) for the special values of the parameters ${\alpha }_1,{\alpha }_2,{\alpha }_3$:

$${\alpha }_1=\mathsf{a}+2k,{\alpha }_2=-2k,{\alpha }_3=-2m,m,k\in {\mathbb{Z}}_{+},$$

(20)

where the symbol ${\mathbb{Z}}_{+}$ contains positive integers and zero. In the notation of the four-component parameters ${\alpha }_i,i=1,2,3,4$, we have

$${\alpha }_i=2({\displaystyle \frac{\mathsf{a}}{2}}+k,-k,-m,1-{\displaystyle \frac{\mathsf{a}}{2}}+m),$$

with $\mathsf{a}$ being an arbitrary variable. In the next few subsections, for the values of the parameters given in (20), we will be able to obtain the solutions of Equation (2) by acting with the translation operators on a class of the seed solutions.

We have already established a connection of Equation (2) to the Painlevé V equation that can be recovered by eliminating one of the two variables. For example, eliminating G from Equation (2), we obtain for $y=F{(F-1)}^{-1}$, the Painlevé V Equation (1), which for the values listed in (20) has the parameters:

$$\alpha ={\displaystyle \frac{1}{2}}{\left(m\right)}^2,\beta =-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\mathsf{a}}{2}}+k\right)}^2,\gamma =1-{\displaystyle \frac{\mathsf{a}}{2}}+m+k,m,k\in {\mathbb{Z}}_{+},$$

that agrees with one of the conditions for the existence of the rational solutions given in [19]. Eliminating instead F from Equation (2), we find that $y=G{(G-1)}^{-1}$ satisfies the Painlevé V Equation (1) with the coefficients:

$$\alpha ={\displaystyle \frac{1}{8}}{\alpha }_4^2,\beta =-{\displaystyle \frac{1}{8}}{\alpha }_2^2,\gamma ={\displaystyle \frac{{\alpha }_1-{\alpha }_3}{2}},\delta =-{\displaystyle \frac{1}{2}}.$$

4.1. Seed Solutions and Translation Operators

There are essentially two fundamental seed solutions to Equation (2):

$$\begin{array}{cc}\hfill F& ={\displaystyle \frac{1}{2}},G={\displaystyle \frac{1}{2}},(\mathsf{a},1-\mathsf{a},\mathsf{a},1-\mathsf{a}),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill F& =1,G=0,(\mathsf{a},0,0,2-\mathsf{a})\hfill \end{array}$$

(22)

with an arbitrary constant $\mathsf{a}$.

The first of these seed solutions given in (21) is invariant under ${\pi }^2$. This class of seed solutions gives rise to Umemura polynomials and is not associated with degeneracy. It is illustrative to recall [11] that in the setting of the even dressing chain (7), the seed solution (21) is represented by solution $j_i\left(z\right)=(z/4)(1,1,1,1)$ with only positive components, while the seed solution (22) is represented by solution $j_i\left(z\right)=(z/4)(1,1,-1,1)$ with one negative and three positive components. In this presentation, we will only study solutions (22) to obtain the closed expressions for the special function solutions generated from it by the Bäcklund transformations. Acting with $\pi$ will generate from the solution (22) equivalent solutions:

$$\begin{array}{cc}\hfill F& =1,G=1,(2-\mathsf{a},\mathsf{a},0,0),\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill F& =0,G=1,(0,2-\mathsf{a},\mathsf{a},0),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill F& =0,G=0,(0,0,2-\mathsf{a},\mathsf{a},),\hfill \end{array}$$

(25)

which, therefore, do not require a separate treatment. We will refer to the seed solution (22) using the following (physics-inspired) notation:

$${|F=1,G=0\rangle }_{(\mathsf{a},0,0,2-\mathsf{a})}.$$

(26)

Within the $A_3^{\left(1\right)}$ extended affine Weyl group, one defines an abelian subgroup of translation operators defined as $T_i=r_{i+3}{r}_{i+2}{r}_{i+1}{r}_i,i=1,2,3,4$, where $r_i=r_{4+i}=s_i$ for $i=1,2,3$ and $r_4=\pi$. The translation operators commute among themselves, $T_i{T}_j=T_j{T}_i$, and generate the following translations when acting on the ${\alpha }_i$ parameters:

$$T_i\left({\alpha }_i\right)={\alpha }_i+2,T_i\left({\alpha }_{i-1}\right)={\alpha }_{i-1}-2,T_i\left({\alpha }_j\right)={\alpha }_j,j=i+1,j=i+2.$$

The inverse translation operators are defined as $T_i^{-1}=r_i^{-1}{r}_{i+1}^{-1}{r}_{i+2}^{-1}{r}_{i+3}^{-1}$, with $r_i^{-1}=r_{4+i}^{-1}=s_i$ for $i=1,2,3$ and $r_4^{-1}={\pi }^{-1}$ and generate the following ${\alpha }_i$ shifts

$$T_i^{-1}\left({\alpha }_i\right)={\alpha }_i-2,T_i^{-1}\left({\alpha }_{i-1}\right)={\alpha }_{i-1}+2,T_i^{-1}\left({\alpha }_j\right)={\alpha }_j,j=i+1,j=i+2.$$

We notice that the Bäcklund transformations $s_2,s_3$ generate infinities when applied on the solution (22) because of $G=0$ and $1-F=0$. To avoid these singularities, only the following solutions are permitted to be generated out of the the seed solution (22) by use of translations [11]:

$$T_1^n{T}_2^{-k}{T}_4^m{|F=1,G=0\rangle }_{{\alpha }_{\mathsf{a}}},n\in \mathbb{Z},k,m\in {\mathbb{Z}}_{+},{\alpha }_{\mathsf{a}}=(\mathsf{a},0,0,2-\mathsf{a}).$$

In this way, we obtain, by action of the translation operators, new solutions $T_1^n{T}_2^{-k}{T}_4^m(F=1)$ and $T_1^n{T}_2^{-k}{T}_4^m(G=0)$ of Equation (2) with new parameters:

$$T_1^n{T}_2^{-k}{T}_4^m(\mathsf{a},0,0,2-\mathsf{a})=(\mathsf{a}+2n+2k,-2k,-2m,2-\mathsf{a}-2n+2m).$$

(27)

One notices that the action of $T_1^n$ merely produces a shift of a parameter $\mathsf{a}$ and, as shown in [11], leaves the configuration $F=1,G=0$ unchanged:

$$T_1^n{|F=1,G=0\rangle }_{{\alpha }_{\mathsf{a}}}={|F=1,G=0\rangle }_{{\alpha }_{\mathsf{a}+2n}}.$$

(28)

We can, therefore, replace the action of $T_1$ by appropriately redefining $\mathsf{a}$ and accordingly restrict our discussion to the solutions of the $F,G$ coupled equations of the form:

$$\begin{array}{cc}\hfill & \mathbb{T}(k,m;\mathsf{a})=T_2^{-k}{T}_4^m{|F=1,G=0\rangle }_{{\alpha }_{\mathsf{a}}},k,m\in {\mathbb{Z}}_{+},\hfill \\ \hfill & {\alpha }_{k,m,\mathsf{a}}=T_2^{-k}{T}_4^m\left({\alpha }_{\mathsf{a}}\right)=(\mathsf{a}+2k,-2k,-2m,2-\mathsf{a}+2m),\hfill \end{array}$$

(29)

where we listed both the solution and its corresponding parameter ${\alpha }_{k,m,\mathsf{a}}$ generated by the combined actions of the translation operators $T_2^{-k}$ and $T_4^m$, with $k,m$ being positive integers, acting on the seed solution (22). In Section 5, we will find closed expressions for the solutions $\mathbb{T}(k,m;\mathsf{a})$ introduced in Relation (29) in terms of the Kummer polynomials.

4.2. Action of Automorphisms for the $N=4$ Model

Taking a look at the $N=4$ Equation (3), we can easily see that they are manifestly invariant under actions generated by the four (two even and two odd) automorphisms $P_i,i=1,2,3,4$:

$$\begin{array}{cc}\hfill P_4& \to :x\to -x,g_1\leftrightarrow g_3,{\alpha }_1\leftrightarrow {\alpha }_3\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill P_2& \to :x\to -x,g_2\leftrightarrow g_4,{\alpha }_2\leftrightarrow {\alpha }_4\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill P_1& \to :x\to -x,g_1\leftrightarrow g_4,g_2\leftrightarrow g_3,{\alpha }_1\leftrightarrow {\alpha }_4,{\alpha }_2\leftrightarrow {\alpha }_3\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill P_3& \to :x\to -x,g_1\leftrightarrow g_2,g_3\leftrightarrow g_4,{\alpha }_1\leftrightarrow {\alpha }_2,{\alpha }_3\leftrightarrow {\alpha }_4,\hfill \end{array}$$

(33)

The above automorphisms follow from the general definitions (11) and (12) valid for an arbitrary even N. They are associated with reflections on a square with vertices $1,2,3,4$. The even reflections $P_2,P_4$ introduced in Relation (11) act as reflections around the diagonals passing through the points 1 and 3 and points 2 and 4, respectively. They act, therefore, as simple transpositions of two points. The automorphisms $P_1,P_3$ are odd reflections introduced in (12). Their corresponding reflection axes are lines going through the centers of two opposite sites of the square.

For $P_1$, these opposite sites are 1–4 and 2–3, for $P_3$, these opposite sites are 1–2 and 3–4.

They all act on ${\alpha }_k,g_k$ via the Relation (13).

A few obvious identities:

$$P_{2i}{P}_{2i+1}=P_{2i+2}{P}_{2i+3},{\widehat{P}}_{2i+1}{\widehat{P}}_2{\widehat{P}}_{2i+1}={\widehat{P}}_4,{\widehat{P}}_{2i}{\widehat{P}}_1{\widehat{P}}_{2i}={\widehat{P}}_3,$$

etc., follow from the basic definitions.

Furthermore, the product of all four reflections ${\widehat{P}}_4{\widehat{P}}_2{\widehat{P}}_1{\widehat{P}}_3$ is an identity due to

$$P_4{P}_2{P}_1{P}_3\left(k\right)=-(2-(1-(3-k))=k-4=k.$$

4.3. Conjugations of the Bäcklund and Translations Operators by Reflections

Conjugation of the Bäcklund transformations by the reflections is described by the general Formula (19) adapted to the $N=4$ case:

$${\widehat{P}}_n{s}_m{\widehat{P}}_n=s_{P_n\left(m\right)},n=2i,2i+1,n,m=1,2,3,4$$

(34)

where $P_n\left(m\right)$ is the point obtained from m by acting with the reflection $P_n$. Repeated conjugation yields:

$${\widehat{P}}_{n+1}{\widehat{P}}_n{s}_m{\widehat{P}}_n{\widehat{P}}_{n+1}=s_{P_{n+1}{P}_n\left(m\right)}=s_{m+1}$$

which, in view of the Relation (15), reproduces a basic relation $\pi s_m=s_{m+1}\pi$.

We are able to extend the above results to include conjugations by reflections of the translation operators. They are described by the following two relations:

$${\widehat{P}}_n{T}_i^{-1}{\widehat{P}}_n=T_{P_{n+1}\left(i\right)},{\widehat{P}}_n{T}_i{\widehat{P}}_n=T_{P_{n+1}\left(i\right)}^{-1}.$$

(35)

Several relevant $N=4$ examples that follow from the above identity are listed below:

$$\begin{array}{cc}\hfill {\widehat{P}}_2{T}_2^{-1}{\widehat{P}}_2& =T_1,{\widehat{P}}_2{T}_4{\widehat{P}}_2=T_3^{-1},\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill {\widehat{P}}_0{T}_2^{-1}{\widehat{P}}_0& =T_3,{\widehat{P}}_0{T}_4{\widehat{P}}_0=T_1^{-1}.\hfill \end{array}$$

(37)

The transformations of the translation operators by odd reflections are as follows:

$${\widehat{P}}_1{T}_2^{-1}{\widehat{P}}_1=T_4,{\widehat{P}}_1{T}_4{\widehat{P}}_1=T_2^{-1},$$

(38)

for ${\widehat{P}}_1$ reflection and

$${\widehat{P}}_3{T}_2^{-1}{\widehat{P}}_3=T_2,{\widehat{P}}_3{T}_4{\widehat{P}}_3=T_4^{-1},$$

(39)

for the ${\widehat{P}}_3$ reflection. The first of these results, given in Relation (38), is the duality map: $k\leftrightarrow m$ that also maps $x\to -x$ and $\mathsf{a}\to 2-\mathsf{a}$, since ${\alpha }_1\leftrightarrow {\alpha }_4$.

5. Closed Expressions for Solutions on the Orbit of Translation Operators

In this section, we find closed expressions for the solutions $\mathbb{T}(k,m;\mathsf{a})$ introduced in Relation (29). We start by first considering the two simpler cases of the $T_4$ orbit and the $T_2$ orbits.

5.1. The $T_4$ and the $T_2$ Orbits and Kummer Polynomials

The $T_4$ orbit and the $T_2$ orbits included in the definition of (29) are described as follows:

• the $k=0$ case that maintains $G=0$, while it transforms F by $T_4$,
• the $m=0$ case that maintains $F=1$ and transforms G by $T_2$.

The $T_4$ orbit:

$$\mathbb{T}(0,m;\mathsf{a})=T_4^m{|F=1,G=0\rangle }_{{\alpha }_{\mathsf{a}}}=\left(F_m(x,\mathsf{a}),G_m(x,\mathsf{a})=0\right),$$

is governed by the recurrence relation

$$F_m=1+{\displaystyle \frac{xm}{x{F}_{m-1}+\mathsf{a}/2-m}},$$

(40)

obtained from the $T_4$ transformation rule

$$\begin{array}{cc}\hfill T_4\left(F\right)& =1-G+{\displaystyle \frac{{\gamma }_1+{\gamma }_4}{F-{\gamma }_4/(1-G)}},\hfill \\ \hfill T_4\left(G\right)& =F-{\displaystyle \frac{{\gamma }_4}{1-G}}+{\displaystyle \frac{{\gamma }_1+{\gamma }_2+{\gamma }_4}{G-\frac{{\gamma }_1+{\gamma }_4}{F-{\gamma }_4/(1-G)}}}.\hfill \end{array}$$

(41)

The above recurrence relation for $F_m$ has a solution:

$$F_m={\displaystyle \frac{N_m(x,\mathsf{a}+2)}{N_m(x,\mathsf{a})}},$$

(42)

in terms of the Kummer polynomials $N_m(x,\mathsf{a})$:

$$N_n(x,\mathsf{a}+2)=2^n{x}^n{(-x)}^{-\mathsf{a}/2}U(-{\displaystyle \frac{\mathsf{a}}{2}},-{\displaystyle \frac{\mathsf{a}}{2}}+n+1,x),$$

(43)

where

$$U(b,b+n+1,x)=x^{-b}\sum _{s=0}^{s=n}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{s}}\right){\left(b\right)}_s{x}^{-s},$$

(44)

is a Kummer polynomial $U(a,b,x)$ in x of degree n when $a-b+1=-n$, $n\in {\mathbb{Z}}_{+}$ [20]. $U(a,b,x)$ solves Kummer’s equation:

$$x{\displaystyle \frac{d^{2}w}{d^{2}x}}+(b-x){\displaystyle \frac{dw}{dx}}-aw=0.$$

Starting with $N_0(x,\mathsf{a})=1$ and applying the first of the basic two recurrence relations:

$$\begin{array}{cc}\hfill N_{m+1}(x,\mathsf{a})& =2x{N}_m(x,\mathsf{a})+(\mathsf{a}-2)N_m(x,\mathsf{a}-2),\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill 2m{N}_{m-1}(x,\mathsf{a})& =N_m(x,\mathsf{a}+2)-N_m(x,\mathsf{a})={\displaystyle \frac{d{N}_m(x,\mathsf{a})}{dx}}m=0,1,2,\dots ,\hfill \end{array}$$

(46)

that can be derived from Relation (43), one first obtains $N_1(x,a)=2x+a-2$ and eventually arrives at a general expression:

$$N_m(x,\mathsf{a})=\sum _{p=0}^{p=m}\left({\displaystyle \genfrac{}{}{0pt}{}{m}{p}}\right){\left(2x\right)}^p(\mathsf{a}-2)(\mathsf{a}-4)\cdots (\mathsf{a}-2(m-p)).$$

(47)

For the $T_2^{-1}$ orbit:

$$\mathbb{T}(k,0;\mathsf{a})=T_2^{-k}{|F=1,G=0\rangle }_{{\alpha }_{\mathsf{a}}}={|F=1,G^k\rangle }_{{\alpha }_{\mathsf{a}}},,$$

(48)

we obtain from the expression for the action by $T_2^{-1}$:

$$T_2^{-1}(F,G)=\left(G-{\displaystyle \frac{{\gamma }_1}{F}}+{\displaystyle \frac{1/x-{\gamma }_2}{1-F+\frac{{\gamma }_1+{\gamma }_4}{1-G+\frac{{\gamma }_1}{F}}}},1-F+{\displaystyle \frac{{\gamma }_1+{\gamma }_4}{1-G+\frac{{\gamma }_1}{F}}}\right),$$

(49)

a chain of transformations that results in a recurrence relation:

$$G^k={\displaystyle \frac{2k}{2x(1-G^{k-1})+(\mathsf{a}+2k-2)}}.$$

(50)

The solution is given this time by

$$G_k=1-{\displaystyle \frac{R_k(x,\mathsf{a}-2)}{R_k(x,\mathsf{a})}}={\displaystyle \frac{2k{R}_{k-1}(x,\mathsf{a})}{R_k(x,\mathsf{a})}},$$

(51)

in terms of the Kummer polynomials:

$$\begin{array}{cc}\hfill & R_n(x,\mathsf{a})=\sum _{p=0}^{p=n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{p}}\right){\left(2x\right)}^p\mathsf{a}(\mathsf{a}+2)(\mathsf{a}+4)\cdots (\mathsf{a}+2(n-p-1))\hfill \\ \hfill & =2^n{x}^n{x}^{\mathsf{a}/2}U({\displaystyle \frac{\mathsf{a}}{2}},{\displaystyle \frac{\mathsf{a}}{2}}+n+1,x),\hfill \end{array}$$

(52)

The above expression can alternatively be obtained from the initial condition $R_0(x,\mathsf{a})=1$ by applying the recursion Relation (53) from the two basic recursive relations satisfied by the polynomials $R_k(x,\mathsf{a})$:

$$\begin{array}{cc}\hfill R_{k+1}(x,\mathsf{a})& =2x{R}_k(x,\mathsf{a})+\mathsf{a}{R}_k(x,\mathsf{a}+2),\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill 2k{R}_{k-1}(x,\mathsf{a})& =R_k(x,\mathsf{a})-R_k(x,\mathsf{a}-2)={\displaystyle \frac{d{R}_k(x,\mathsf{a})}{dx}}.\hfill \end{array}$$

(54)

The polynomials denoted by $N_k(x,\mathsf{a})$ for the $T_4$-orbit and $R_k(x,\mathsf{a})$ for the $T_2$-orbit are both of order k, and are related to each other through the relation:

$$R_k(-x,\mathsf{a})={(-1)}^k{N}_k(x,2-\mathsf{a}).$$

(55)

that is consistent with the $P_1$ reflection that induces $x\to -x,F\to 1-G,G\to 1-F$ and ${\alpha }_1\leftrightarrow {\alpha }_4,{\alpha }_2\leftrightarrow {\alpha }_3$. For ${\alpha }_i$ given in Relation (20), this is equivalent to

$${\widehat{P}}_1:k\leftrightarrow m,\mathsf{a}\leftrightarrow 2-\mathsf{a},x\leftrightarrow -x.$$

(56)

5.2. A General Case of $T_4^m{T}_2^{-k}$ Orbit and the Generalized Higher-Type Kummer Polynomials

We now consider the full orbit given as in (29) by:

$$T_2^{-k}{T}_4^m{|F=1,G=0\rangle }_{{\alpha }_{\mathsf{a}}}=\left(F_m^k(x,\mathsf{a}),G_m^k(x,\mathsf{a})\right),k,m\in {\mathbb{Z}}_{+}.$$

Here, $F_m^k(x,\mathsf{a}),G_m^k(x,\mathsf{a})$ are solutions to Equation (2)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dx}}{F}_m^k& =F_m^k(F_m^k-1)(2G_m^k-1)-{\displaystyle \frac{\mathsf{a}+2k-2m}{2x}}{F}_m^k+{\displaystyle \frac{\mathsf{a}+2k}{2x}},\hfill \\ \hfill {\displaystyle \frac{d}{dx}}{G}_m^k& =-G_m^k(G_m^k-1)(2F_m^k-1)+{\displaystyle \frac{\mathsf{a}+2k-2m-2}{2x}}{G}_m^k+{\displaystyle \frac{-2k}{2x}},\hfill \end{array}$$

(57)

with parameters given in (20).

See also reference [21,22] for use of the translation operators to obtain the solution of the Painlevé V equation in terms of the Laguerre polynomials and [23] for a more recent discussion.

Here, we are able to find the Kummer polynomial representation for both $F_m^k$ and $G_m^k$, as given in the main result of this section by the following expression:

$$\begin{array}{cc}\hfill F_m^k(x,\mathsf{a})& ={\displaystyle \frac{R_{mk}^{(m-1)}(x,\mathsf{a}-2)R_{m(k+1)}^{(m-1)}(x,\mathsf{a})}{R_{mk}^{(m-1)}(x,\mathsf{a})R_{m(k+1)}^{(m-1)}(x,\mathsf{a}-2)}},\hfill \\ \hfill G_m^k(x,\mathsf{a})& =2k{\displaystyle \frac{R_{m(k+1)}^{(m-1)}(x,\mathsf{a}-2)R_{(m+1)(k-1)}^{\left(m\right)}(x,\mathsf{a})}{R_{mk}^{(m-1)}(x,\mathsf{a}-2)R_{(m+1)k}^{\left(m\right)}(x,\mathsf{a})}},m,k=0,1,2,\dots .\hfill \end{array}$$

(58)

The symbols $R_n^{\left(0\right)}(x,\mathsf{a})=R_n(x,\mathsf{a})$ are regular Kummer polynomials of the n-th order defined in Relation (52). The symbols $R_n^{\left(m\right)}(x,\mathsf{a})$ are generalized Kummer polynomials of the n-th order of the m-th type. They are obtained from the Kummer polynomials by a set of recurrence relations:

$$\begin{array}{cc}\hfill R_{(m-1)(k+1)}^{(m-2)}(x,\mathsf{a}-2)R_{(m+1)k}^{\left(m\right)}(x,\mathsf{a})& ={\displaystyle \frac{1}{2m}}[R_{mk}^{(m-1)}(x,\mathsf{a}-2)R_{m(k+1)}^{(m-1)}(x,\mathsf{a})\hfill \\ & -R_{mk}^{(m-1)}(x,\mathsf{a})R_{m(k+1)}^{(m-1)}(x,\mathsf{a}-2)],\hfill \end{array}$$

(59)

that are valid for $m=1,2,3,\dots$ and $k=0,1,\dots$, with the initial variables being $R_0^{(-1)}(x,\mathsf{a}-2)=1$ and $R_k^{\left(0\right)}(x,\mathsf{a})=R_k(x,\mathsf{a})$. The master recurrence Relation (59) gives rise to the determinant expressions for $R_{mk}^{(m-1)}(x,\mathsf{a})$, which we will refer to as $(m-1)$-type Kummer-like polynomials of order $mk$. Their determinant expressions are obtained by employing the Desnanot–Jacobi identity:

$$\left|M\right|\times |M_{1,n}^{1,n}|=|M_n^n|\times |M_1^1|-|M_1^n|\times |M_n^1|,$$

(60)

which is valid for any a $(k+1)\times (k+1)$ matrix M. The notation in identity (60) is such that $M_i^j$ is a matrix M with the j-th row and i-th column removed. When the Desnanot–Jacobi identity is applied on the above recurrence Relation (59) (up to a constant $C_{m-1}$ determined below), one finds $mk$ order and the $(m-1)$-type Kummer-like polynomial $R_{mk}^{(m-1)}(x,\mathsf{a})$ being the determinant of the $m\times m$ matrix M with the matrix elements to be given by:

$$R_{mk}^{(m-1)}(x,\mathsf{a})=C_{m-1}\left|M\right|,M_{ij}=R_{k+i-1}(x,\mathsf{a}-2(m-j)),i,j=1,\dots ,m,$$

(61)

where $R_n(x,b)$ are the original Kummer polynomials (52) and the constants satisfy the relation:

$$C_m={\displaystyle \frac{1}{2m}}{\displaystyle \frac{C_{m-1}^2}{C_{m-2}}},m=2,3,4,\dots .$$

(62)

It follows for the first few cases that

$$C_0=1,C_1={\displaystyle \frac{1}{2}},C_2={\displaystyle \frac{1}{2·2}}{\left({\displaystyle \frac{1}{2}}\right)}^2={\displaystyle \frac{1}{16}}.$$

Generally, we find from Relation (62):

$$C_m={\displaystyle \frac{1}{2^{m(m+1)/2}}}\prod _{j=0}^{m-2}{\displaystyle \frac{1}{{(m-j)}^{j+1}}},m=2,3,4,\dots$$

(63)

The value of $C_m$ is such that the identity $R_0^{\left(m\right)}(x,a)=1,m=0,1,2,\dots$ holds as follows from the above determinant expressions.

From the determinant expression (61), we find as a special example, $R_{2k}^{\left(1\right)}(x,\mathsf{a})$ to be given by:

$$\begin{array}{cc}\hfill R_{2k}^{\left(1\right)}(x,\mathsf{a})& ={\displaystyle \frac{1}{2}}\left|\begin{array}{cc}{R}_k(x,\mathsf{a}-2)& R_k(x,\mathsf{a})\\ R_{k+1}(x,\mathsf{a}-2)& R_{k+1}(x,\mathsf{a})\end{array}\right|\hfill \\ & ={\displaystyle \frac{1}{2}}\left(R_2(x,\mathsf{a})R_1(x,\mathsf{a}-2)-R_1(x,\mathsf{a})R_2(x,\mathsf{a}-2)\right).\hfill \end{array}$$

(64)

One can check explicitly that

$$R_2^{\left(1\right)}(x,\mathsf{a})=N_2(x,\mathsf{a}+2),$$

(65)

which is a special case of the “duality” Relation (71) to be introduced below and shown to play an important role in a general scheme illustrating the power of symmetry generated by the ${\widehat{P}}_1$ automorphism.

Other basic examples of $R_{mk}^{(m-1)}$ are given by the following determinant expressions for $m=3$ and $m=4$:

$$\begin{array}{cc}\hfill R_{3k}^{\left(2\right)}(x,\mathsf{a})& ={\displaystyle \frac{1}{16}}\left|\begin{array}{ccc}{R}_k(x,\mathsf{a}-4)& R_k(x,\mathsf{a}-2)& R_k(x,\mathsf{a})\\ R_{k+1}(x,\mathsf{a}-4)& R_{k+1}(x,\mathsf{a}-2)& R_{k+1}(x,\mathsf{a})\\ R_{k+2}(x,\mathsf{a}-4)& R_{k+2}(x,\mathsf{a}-2)& R_{k+2}(x,\mathsf{a})\end{array}\right|k=1,2,3,\dots ,\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill R_{4k}^{\left(3\right)}(x,\mathsf{a})& ={\displaystyle \frac{1}{3}}{\left({\displaystyle \frac{1}{16}}\right)}^2\left|\begin{array}{cccc}{R}_k(x,\mathsf{a}-2·3)& R_k(x,\mathsf{a}-2·2)& R_k(x,\mathsf{a}-2)& R_k(x,\mathsf{a})\\ R_{k+1}(x,\mathsf{a}-2·3)& R_{k+1}(x,\mathsf{a}-2·2)& R_{k+1}(x,\mathsf{a}-2)& R_{k+1}(x,\mathsf{a})\\ R_{k+2}(x,\mathsf{a}-2·3)& R_{k+2}(x,\mathsf{a}-2·2)& R_{k+2}(x,\mathsf{a}-2)& R_{k+2}(x,\mathsf{a})\\ R_{k+3}(x,\mathsf{a}-2·3)& R_{k+3}(x,\mathsf{a}-2·2)& R_{k+3}(x,\mathsf{a}-2)& R_{k+3}(x,\mathsf{a})\end{array}\right|.\hfill \end{array}$$

(67)

It follows from the above determinant expressions that the Kummer polynomials of the p-th kind and of the n-th order are described by a general formula:

$$R_n^{\left(p\right)}(x,\mathsf{a})=\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\left(2x\right)}^k\left(\mathsf{a}-a_1^{(n,k,p)}\right)\dots \left(\mathsf{a}-a_{n-k}^{(n,k,p)}\right),p\ge 0,n>0,R_0^{\left(p\right)}(x,\mathsf{a})=1,$$

(68)

and thus are fully defined by the roots $a_i^{(n,k,p)}$ of the monomials for $i=1,\dots ,n-k$.

Using equation: $d{R}_n(x,\mathsf{a})/dx=R_n(x,\mathsf{a})-R_n(x,\mathsf{a}-2)$, we find that

$$\begin{array}{cc}\hfill R_n(x,\mathsf{a}-6)& =R_n(x,\mathsf{a})-3R_n^{\prime }(x,\mathsf{a})+3R_n^{″}(x,\mathsf{a})-R_n^{‴}(x,\mathsf{a}),\hfill \\ \hfill R_n(x,\mathsf{a}-4)& =R_n(x,\mathsf{a})-2R_n^{\prime }(x,\mathsf{a})+R_n^{″}(x,\mathsf{a}),R_n(x,\mathsf{a}-2)=R_n(x,\mathsf{a})-R_n^{\prime }(x,\mathsf{a}),\hfill \end{array}$$

and accordingly, the determinant in Equation (67) can be rewritten as the Wronskian of the Kummer polynomials:

$$R_{4k}^{\left(3\right)}(x,\mathsf{a})={\displaystyle \frac{1}{3}}{\left({\displaystyle \frac{1}{16}}\right)}^2\left|\begin{array}{cccc}{R}_k^{‴}(x,\mathsf{a})& R_k^{″}(x,\mathsf{a})& R_k^{\prime }(x,\mathsf{a})& R_k(x,\mathsf{a})\\ R_{k+1}^{‴}(x,\mathsf{a})& R_{k+1}^{″}(x,\mathsf{a})& R_{k+1}^{\prime }(x,\mathsf{a})& R_{k+1}(x,\mathsf{a})\\ R_{k+2}^{‴}(x,\mathsf{a})& R_{k+2}^{″}(x,\mathsf{a})& R_{k+2}^{\prime }(x,\mathsf{a})& R_{k+2}(x,\mathsf{a})\\ R_{k+3}^{‴}(x,\mathsf{a})& R_{k+3}^{″}(x,\mathsf{a})& R_{k+3}^{\prime }(x,\mathsf{a})& R_{k+3}(x,\mathsf{a})\end{array}\right|,$$

(69)

with all polynomials now taken at the same point $\mathsf{a}$.

There also exists a dual master recurrence relation:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2k}}\left(R_{mk}^{(m-1)}(x,\mathsf{a}-2)R_{(m+1)k}^{\left(m\right)}(x,\mathsf{a})-R_{mk}^{(m-1)}(x,\mathsf{a})R_{k(m+1)}^{\left(m\right)}(x,\mathsf{a}-2)\right)\hfill \\ & =R_{m(k+1)}^{(m-1)}(x,\mathsf{a}-2)R_{(k-1)(m+1)}^{\left(m\right)}(x,\mathsf{a}).\hfill \end{array}$$

(70)

Applying the duality relation

$$R_{km}^{(m-1)}(x,\mathsf{a}-2)=N_{km}^{(k-1)}(x,\mathsf{a}),$$

(71)

that exchanges $m\leftrightarrow k$ and $\mathsf{a}\to \mathsf{a}-2$ and introduces a higher- type order generalization of the Kummer polynomials $N_k(x,\mathsf{a})$ defined in (43), we obtain the dual version of the recurrence Relation (59):

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2k}}\left(N_{mk}^{(k-1)}(x,\mathsf{a})N_{(m+1)k}^{(k-1)}(x,\mathsf{a}+2)-N_{mk}^{(k-1)}(x,\mathsf{a}+2)N_{k(m+1)}^{(k-1)}(x,\mathsf{a})\right)\hfill \\ & =N_{m(k+1)}^{\left(k\right)}(x,\mathsf{a})N_{(k-1)(m+1)}^{(k-2)}(x,\mathsf{a}+2),\hfill \end{array}$$

(72)

that via the Desnanot–Jacobi identity (60) leads to an expression for the generalized higher-type Kummer polynomials $N_k(x,\mathsf{a})$:

$$N_{km}^{(k-1)}(x,\mathsf{a})=C_{k-1}\left|M\right|,M_{ij}=N_{m+i-1}(x,\mathsf{a}+2(j-1)),$$

(73)

where the normalization constant $C_{k-1}$ is defined in (62). For example,

$$N_{3m}^{\left(2\right)}(x,(x,\mathsf{a}))={\displaystyle \frac{1}{16}}\left|\begin{array}{ccc}{N}_m(x,\mathsf{a})& N_m(x,\mathsf{a}+2)& N_m(x,\mathsf{a}+4)\\ N_{m+1}(x,\mathsf{a})& N_{m+1}(x,\mathsf{a}+2)& N_{m+1}(x,\mathsf{a}+4)\\ N_{m+2}(x,\mathsf{a})& N_{m+2}(x,\mathsf{a}+2)& N_{m+2}(x,\mathsf{a}+4)\end{array}\right|m=1,2,3,\dots$$

(74)

Thus, we have the determinant expressions (61) and (73) that are dual to each other according to the duality Relation (71) that can also be viewed as generated by the ${\widehat{P}}_1$ transformation $m\leftrightarrow k$, $x\to -x,\mathsf{a}\to 2-\mathsf{a}$. In view of the duality Relation (71), we have the following relation:

$$R_{km}^{(m-1)}(-x,-\mathsf{a})={(-1)}^{km}{R}_{km}^{(k-1)}(x,\mathsf{a}).$$

(75)

To understand better the Relation (75), it is illustrative to introduce the notation $R_{m,k}(x,\mathsf{a})=R_{mk}^{(m-1)}(x,\mathsf{a})$. In this new notation, the Relation (75) describes a condition for the commutation of the indices $m,k\to k,m$: $R_{k,m}(x,\mathsf{a})={(-1)}^{km}{R}_{m,k}(-x,-\mathsf{a})$.

Also, thanks to the duality, we have an alternative expression for our solutions as follows:

$$\begin{array}{cc}\hfill F_m^k(x,\mathsf{a})& ={\displaystyle \frac{N_{mk}^{(k-1)}(x,\mathsf{a})N_{m(k+1)}^{\left(k\right)}(x,\mathsf{a}+2)}{N_{mk}^{(k-1)}(x,\mathsf{a}+2)N_{m(k+1)}^{\left(k\right)}(x,\mathsf{a})}},\hfill \\ \hfill G_m^k(x,\mathsf{a})& =2k{\displaystyle \frac{N_{m(k+1)}^{\left(k\right)}(x,\mathsf{a})N_{(k-1)(m+1)}^{(k-2)}(x,\mathsf{a}+2)}{N_{mk}^{(k-1)}(x,\mathsf{a})N_{(m+1)k}^{(k-1)}(x,\mathsf{a}+2)}}.\hfill \end{array}$$

(76)

6. Even ${\widehat{\mathit{P}}}_{\mathbf{2}},{\widehat{\mathit{P}}}_{\mathbf{4}}$ Reflections and Degeneracy

A general remark that applies to the automorphisms ${\widehat{P}}_4,{\widehat{P}}_2,{\widehat{P}}_3$ is that they transform the integers k and m into the sum and differences of $k,m$ and $\mathsf{a}/2$. Thus, consistently, to have integers of both sides of the equations requires that the constant $\mathsf{a}/2$ be an integer, chosen here to be $l\in \mathbb{Z}$.

6.1. Action of the ${\widehat{P}}_2$ Automorphism

For ${\widehat{P}}_2$, the transformation ${\alpha }_2\leftrightarrow {\alpha }_4$ for ${\alpha }_2=-2k$ and ${\alpha }_4=2-\mathsf{a}+2m=2(1-l+m)$ is equivalent to that of

$$l\to 1+m+k,k\to l-1-m,$$

for $l>m$.

Accordingly, the ${\widehat{P}}_2$ transformations are such that

$$\begin{array}{cc}\hfill {\widehat{P}}_2\left(F_m^k(x,l)\right)& =F_m^{l-1-m}(-x,1+m+k)\hfill \\ \hfill {\widehat{P}}_2\left(G_m^k(x,l)\right)& =G_m^{l-1-m}(-x,1+m+k),\mathsf{a}=2l\hfill \end{array}$$

and are expected to be solutions to Equation (2) with the same parameters as the solutions $F_m^k(x,\mathsf{a}),1-G_m^k(x,\mathsf{a})$. Are these functions equal to the original solutions $F_m^k(x,\mathsf{a}),1-G_m^k(x,\mathsf{a})$ or are they degenerated solutions is the main question. We will fully answer it in Section 6.3, but first, we will discuss how it works on an explicit example:

Example 1. 

Here, we calculate a solution $T_2^{-k}{T}_4^m{|F=1,G=0\rangle }_{{\alpha }_{\mathsf{a}=4}}$ using the expression for solutions found in Equation (58) for $k=m=1$ with $l=2$ ($\mathsf{a}=2l=4$):

$$\begin{array}{cc}\hfill F_1^1(x,\mathsf{a}=4)& ={\displaystyle \frac{R_1(x,\mathsf{a}-2)R_2(x,\mathsf{a})}{R_1(x,\mathsf{a})R_2(x,\mathsf{a}-2)}}{|}_{\mathsf{a}=4}={\displaystyle \frac{(2x+2)({\left(2x\right)}^2+16x+24)}{(2x+4)({\left(2x\right)}^2+8x+8)}}\hfill \\ \hfill G_1^1(x,\mathsf{a}=4)& =2{\displaystyle \frac{R_2(x,\mathsf{a}-2)R_0^{\left(1\right)}(x,\mathsf{a})}{R_1(x,\mathsf{a}-2)R_2^{\left(1\right)}(x,\mathsf{a})}}{{|}_{\mathsf{a}=4}=2{\displaystyle \frac{R_2(x,\mathsf{a}-2)}{R_1(x,\mathsf{a}-2)N_2(x,\mathsf{a}+2)}}|}_{\mathsf{a}=4}\hfill \\ \hfill & =2{\displaystyle \frac{({\left(2x\right)}^2+8x+8)}{(2x+2)({\left(2x\right)}^2+16x+8)}},\hfill \end{array}$$

(77)

where we used the identities (65) and $R_0^{\left(m\right)}(x,\mathsf{a})=1$. The above two solutions in Relation (77) satisfy the Equation (57) with $k=m=1$, (${\alpha }_1=\mathsf{a}+2=6,{\alpha }_2=-2,{\alpha }_3=-2$). We write these parameters as follows:

$${\alpha }_i=2(k+l,-k,-m,1-l+m)=2(3,-1,-1,0),$$

(78)

Under the ${\widehat{P}}_2$ transformation, the parameter m is invariant and we have the following expressions for the $\mathsf{a}$, k transformations:

$$\begin{array}{cc}\hfill \overline{\mathsf{a}}& =\mathsf{a}+2(1+k+m-l)=4+2=6\hfill \\ \hfill \overline{k}& =k-(1+k+m-l)=1-1=0\hfill \end{array}$$

Thus, we are looking for the transformed solutions given by:

$$\begin{array}{cc}\hfill F_m^{\overline{k}}(-x,\overline{\mathsf{a}}/2)& =F_1^0(-x,6),\hfill \\ \hfill G_m^{\overline{k}}(-x,\overline{\mathsf{a}}/2)& =G_1^0(-x,6),\hfill \end{array}$$

which we calculate using Equation (58) to obtain:

$$\begin{array}{cc}\hfill F_1^0(-x,6)& ={\displaystyle \frac{R_0(-x,4)R_1(-x,6)}{R_0(-x,6)R_1(-x,4)}}={\displaystyle \frac{6-2x}{4-2x}}={\displaystyle \frac{x-3}{x-2}},\hfill \\ \hfill G_1^0(-x,6)& =2·0·{\displaystyle \frac{R_1(-x,4)R_{-2)}^{\left(1\right)}(-x,6)}{R_0(-x,4)R_0^{\left(1\right)}(-x,6)}}=0.\hfill \end{array}$$

(79)

One checks explicitly that $F=(x-3)/(x-2)$ and $G=1-G_0^1(-x,6)=1$ solve the Equation (2) with ${\alpha }_1=6,{\alpha }_2=-2,{\alpha }_3=-2$ that agrees with the parameters (78) of the original solution. We will see in Section 6.3 that this degeneracy is a general feature of the even ${\widehat{P}}_{2i}$ automorphisms.

Generally, we can rewrite

$$\begin{array}{cc}\hfill F_m^{l-1-m}(-x,2(1+m+k))& ={\displaystyle \frac{R_{m(l-1-m)}^{(m-1)}(-x,2(k+m))R_{m(l-m)}^{(m-1)}(-x,2(1+m+k))}{R_{m(l-1-m)}^{(m-1)}(-x,2(1+m+k))R_{m(l-m)}^{(m-1)}(-x,2(k+m))}}\hfill \\ \hfill & ={\displaystyle \frac{N_{m(l-1-m)}^{(m-1)}(x,2(1-k-m))N_{m(l-m)}^{(m-1)}(x,-2(m+k))}{N_{m(l-1-m)}^{(m-1)}(x,-2(m+k))N_{m(l-m)}^{(m-1)}(x,2(1-k-m))}}\hfill \end{array}$$

(80)

where we used the identity

$$R_{mk}^{(m-1)}(x,\mathsf{a})={(-1)}^{mk}{N}_{mk}^{(m-1)}(-x,2-a),R_{mk}^{(m-1)}(-x,a)={(-1)}^{mk}{N}_{mk}^{(m-1)}(x,2-a),$$

(81)

Expression (80) describes a closed expression for the degenerated partner of the following:

$$F_m^k(x,2l)={\displaystyle \frac{R_{mk}^{(m-1)}(x,2(l-1)R_{m(k+1)}^{(m-1)}(x,2l)}{R_{mk}^{(m-1)}(x,2l)R_{m(k+1)}^{(m-1)}(x,2(l-1)}},$$

which is a solution given in (58) for $m,k,a=2l$.

For the remaining part of the solution, we find that the degenerated solution is as follows:

$$\begin{array}{cc}\hfill & G_m^{l-1-m}(-x,2(1+m+k))\hfill \\ & =2(l-1-m){\displaystyle \frac{R_{m(l-m)}^{(m-1)}(-x,2(m+k))R_{(l-m-2)(m+1)}^{\left(m\right)}(-x,2(1+m+k))}{R_{m(l-1-m)}^{(m-1)}(-x,2(m+k))R_{(m+1)(l-1-m)}^{\left(m\right)}(-x,2(1+m+k))}}\hfill \\ \hfill & =-2(l-1-m){\displaystyle \frac{N_{m(l-m)}^{(m-1)}(x,2(1-m-k))N_{(l-m-2)(m+1)}^{\left(m\right)}(x,-2(m+k))}{N_{m(l-1-m)}^{(m-1)}(x,2(1-m-k))N_{(m+1)(l-1-m)}^{\left(m\right)}(x,-2(m+k))}},\hfill \end{array}$$

(82)

where we again used Equation (81).

Remember that ${\widehat{P}}_2\left(G\right)=1-G_m^{l-1-m}(-x,2(1+m+k))$. Thus,

$${\widehat{P}}_2\left(G\right)=1+2(l-1-m){\displaystyle \frac{N_{m(l-m)}^{(m-1)}(x,2(1-m-k))N_{(l-m-2)(m+1)}^{\left(m\right)}(x,-2(m+k))}{N_{m(l-1-m)}^{(m-1)}(x,2(1-m-k))N_{(m+1)(l-1-m)}^{\left(m\right)}(x,-2(m+k))}}.$$

(83)

Together with (80), Formula (83) solves (2) with the identical parameters ${\alpha }_i=2(l+k,-k,-m,1-l+m)$ as the original solution (58) with $\mathsf{a}=2l$.

Example 2. 

We consider the case of

$$k=1,m=1,l=4,a=2·l=8\to {\alpha }_i=2(l+k,-k,-m,1-l+m)=2(5,-1,-1,-2),$$

(84)

We find $\overline{k}=l-1-m=2$, $\overline{l}=(1+m+k)=3$. Applying the Formulas (80) and (83), we obtain the ${\widehat{P}}_2$ transformed degenerated solutions:

$$\begin{array}{cc}\hfill F_1^2(-x,2·3)& ={\displaystyle \frac{N_2(x,-2·1)N_3(x,-2·2)}{N_2(x,-2·2)N_3(x,-2·1)}}\hfill \\ & ={\displaystyle \frac{(x^2-4x+6)(x^3-9x^2+36x-60)}{(x^2-6x+12)(x^3-6x^2+18x-24)}}\hfill \\ \hfill 1-G_1^2(-x,2·3)& =1-(-2·2){\displaystyle \frac{N_3(x,-2))N_2^{\left(1\right)}(x,-4)}{N_2(x,-2)N_4^{\left(1\right)}(x,-4)}}\hfill \\ & ={\displaystyle \frac{(x^2-6x+12)(x^4-8x^3+24x^2-24x+12)}{(x^2-4x+6)(72+54x^2+x^4-12x^3-96x)}}.\hfill \end{array}$$

Inserting $x=-z^2/2$ and multiplying by z, we obtain from the above two results the expressions (3.15) for $q\left(z\right),p\left(z\right)$ of Example 3.1 of [12] that also solve the relevant Painlevé equations with the parameters ${\alpha }_i=2(5,-1,-1,-2)$.

6.2. Action of the ${\widehat{P}}_4$ Automorphism

The ${\widehat{P}}_4$ generated transformation ${\alpha }_1\leftrightarrow {\alpha }_3$ for ${\alpha }_1=2(l+k)$ and ${\alpha }_3=-2m$ is equivalent to that of

$$l\to -k-m,k\to k,m\to -l-k.$$

Thus, it must hold that $-l-k\ge 0$ or $l\le -k$.

Accordingly, we obtain:

$$\begin{array}{cc}\hfill {\widehat{P}}_4\left(F_m^k(x,2l)\right)& =F_{-l-k}^k(-x,2(-k-m)),\hfill \\ \hfill {\widehat{P}}_4\left(G_m^k(x,2l)\right)& =G_{-l-k}^k(-x,2(-k-m))\hfill \end{array}$$

(85)

The first of the Equation (85) yields after substituting the Relation (58) for $G_m^k(x,\mathsf{a})$:

$$\begin{array}{cc}\hfill G_{-l-k}^k(-x,-k-m)& =2k{\displaystyle \frac{R_{(-l-k)(k+1)}^{(-l-k-1)}(-x,2(-k-m-1))R_{(-l-k+1)(k-1)}^{(-l-k)}(-x,2(-k-m))}{R_{(-l-k)k}^{(-l-k-1)}(-x,2(-k-m-1))R_{(-l-k+1)k}^{(-l-k)}(-x,2(-k-m))}}\hfill \\ & =-2k{\displaystyle \frac{N_{(-l-k)(k+1)}^{(-l-k-1)}(x,2(2+k+m))N_{(-l-k+1)(k-1)}^{(-l-k)}(x,2(1+k+m))}{N_{(-l-k)k}^{(-l-k-1)}(x,2(2+k+m))N_{(-l-k+1)k}^{(-l-k)}(x,2(1+k+m))}}\hfill \end{array}$$

From ${\widehat{P}}_4\left(F\right)=F_{-l-k}^k(-x,-2(k+m))$, we obtain

$$\begin{array}{cc}\hfill F_{-l-k}^k(-x,-2(k+m))& ={\displaystyle \frac{R_{(-(l-k)k}^{(-l-k-1)}(-x,-2(1+k+m))R_{(-l-k)(k+1)}^{(-l-k-1)}(-x,-2(k+m))}{R_{(-l-k)k}^{(-l-k-1)}(-x,-2(k+m))R_{(l-k)(k+1)}^{(-l-k-1)}(-x,-2(1+k+m))}}\hfill \\ & ={\displaystyle \frac{N_{(-(l-k)k}^{(-l-k-1)}(x,2(2+k+m))N_{(-l-k)(k+1)}^{(-l-k-1)}(x,2(1+k+m))}{N_{(-l-k)k}^{(-l-k-1)}(x,2(1+k+m))N_{(l-k)(k+1)}^{(-l-k-1)}(x,2(2+k+m))}}\hfill \end{array}$$

We will apply these results in Section 6.3 and Example 4, where we will study the degeneracy introduced by even reflection automorphisms and answer the question whether the results for ${\widehat{P}}_4\left(F_m^k(x,2l)\right)$ and ${\widehat{P}}_4\left(G_m^k(x,2l)\right)$ are equal to $1-F_m^k(x,2l)$ and $G_m^k(x,2l)$ that share the same parameters.

6.3. Degeneracy Induced by Even Reflections ${\widehat{P}}_2$ and ${\widehat{P}}_4$

We now present a general discussion on the ${\widehat{P}}_4,{\widehat{P}}_2$ reflections that acting on solutions $T_2^{-k}{T}_4^m{|g_1=1,g_2=0\rangle }_a$ produce degenerated solutions (solutions that share the same parameters ${\alpha }_i$ as the original solutions).

The result will follow if we are able to show the following equalities:

$$\begin{array}{c}\hfill {\widehat{P}}_2{T}_2^{-k_3}{T}_4^{m_3}{|g_1=1,g_2=0\rangle }_{a_3}=\pi s_1{T}_2^{-k_1}{T}_4^{m_1}{|g_1=1,g_2=0\rangle }_{a_1},\end{array}$$

(86)

$$\begin{array}{c}\hfill {\widehat{P}}_4{T}_2^{-k_4}{T}_4^{m_4}{|g_1=1,g_2=0\rangle }_{a_4}={\pi }^{-1}{s}_4{T}_2^{-k_2}{T}_4^{m_2}{|g_1=1,g_2=0\rangle }_{a_2},\end{array}$$

(87)

for some values of $k_i,m_i$ and $a_i$, since we have previously established in [12] that both solutions on the right-hand sides are different from $T_2^{-k}{T}_4^m{|g_1=1,g_2=0\rangle }_a$, while they share the identical parameters.

First, we will show that the two relations given in Equations (86) and (87) are equivalent. Assume that Equation (86) holds and multiply the Equation (86) by ${\widehat{P}}_2$. Due to relation $\pi ={\widehat{P}}_2{\widehat{P}}_1$, we obtain that Equation (86) is equivalent to:

$$\begin{array}{cc}\hfill T_2^{-k_3}{T}_4^{m_3}{|g_1=1,g_2=0\rangle }_{a_3}& ={\widehat{P}}_1{s}_1{T}_2^{-k_1}{T}_4^{m_1}{|g_1=1,g_2=0\rangle }_{a_1}\hfill \\ & =s_4{\widehat{P}}_1{T}_2^{-k_1}{T}_4^{m_1}{|g_1=1,g_2=0\rangle }_{a_1}\hfill \end{array}$$

where in the last equation, we used that ${\widehat{P}}_1{s}_1{\widehat{P}}_1=s_4$. Multiplying both sides by $s_4$ and then using that ${\widehat{P}}_1{\widehat{P}}_2={\pi }^{-1}$, we obtain:

$$\begin{array}{cc}\hfill {\pi }^{-1}{s}_4{\widehat{P}}_2{T}_2^{-k_3}{T}_4^{m_3}{|g_1=1,g_2=0\rangle }_{a_3}& ={\widehat{P}}_1{\widehat{P}}_2{\widehat{P}}_1{s}_1{T}_2^{-k_1}{T}_4^{m_1}{|g_1=1,g_2=0\rangle }_{a_1}\hfill \\ & ={\widehat{P}}_4{T}_2^{-k_1}{T}_4^{m_1}{|g_1=1,g_2=0\rangle }_{a_1},\hfill \end{array}$$

where we used the identity ${\widehat{P}}_1{\widehat{P}}_2{\widehat{P}}_1={\widehat{P}}_4$ that follows from (17). Thus, Equation (87) follows from Equation (86) and we will only need to show that one of these two equations holds.

We now embark on a proof of Equation (86). First, we recall that

$$\begin{array}{cc}\hfill {\widehat{P}}_2|g_1=1,g_3=0,g_2=0,g_4=1\rangle & =|g_1=1,g_3=0,g_4=0,g_2=1\rangle ,\hfill \\ \hfill {\widehat{P}}_2(a,0,0,2-a)& =(a,2-a,0,0),\hfill \end{array}$$

then, using (36), we obtain

$$\begin{array}{cc}\hfill & {\widehat{P}}_2{T}_2^{-k_3}{T}_4^{m_3}{|g_1=1,g_3=0,g_2=0,g_4=1\rangle }_{a_3}\hfill \\ \hfill & =T_1^{k_3}{T}_3^{-m_3}{|g_1=1,g_3=0,g_4=0,g_2=1\rangle }_{(a_3,2-a_3,0,0)}.\hfill \end{array}$$

(88)

Next, we consider the parameters of solution (88) with $k_3,m_3$ and $a_3$:

$$T_1^{k_3}{T}_3^{-m_3}(a_3,2-a_3,0,0)=(a_3+2k_3,2-a_3+2m_3,-2m_3,-2k_3),$$

that will agree with the parameters $(a+2k,-2k,-2m,2-a+2m)$ of the solution $T_2^{-k}{T}_4^m|g_1=1,g_3=0,g_2=0,g_4=1\rangle$ if $a=2l$, $a_3=2l_3$ and if it holds that

$$k_3=l-m-1,l_3=1+m+k,m_3=m.$$

(89)

We will see that for these values we have a degeneracy.

Let us compare the two sides of expressions (86). Since we have the identity $\pi ={\widehat{P}}_2{\widehat{P}}_1$, we can rewrite the Equation (86) as

$${\widehat{P}}_1{s}_1{T}_2^{-k_1}{T}_4^{m_1}|g_1=1,g_2{=0\rangle }_{a_1}=T_2^{-k_3}{T}_4^{m_3}{|g_1=1,g_2=0\rangle }_{a_3},$$

or

$$s_1{T}_2^{-k_1}{T}_4^{m_1}|g_1=1,g_2{=0\rangle }_{a_1}={\widehat{P}}_1{T}_2^{-k_3}{T}_4^{m_3}|g_1=1,g_2{=0\rangle }_{a_3}=T_4^{k_3}{T}_2^{-m_3}{\widehat{P}}_1{|g_1=1,g_2=0\rangle }_{a_3}.$$

(90)

The left-hand side of Equation (90) can be rewritten as

$$\begin{array}{cc}\hfill & T_1^{-k_1}{T}_4^{m_1}{s}_1|g_1=1,g_2{=0\rangle }_{a_1}=T_1^{-k_1}{T}_4^{m_1}{s}_1{T}_1^{a_1/2}{|g_1=1,g_2=0\rangle }_{a=0}\hfill \\ \hfill & =T_1^{-k_1}{T}_4^{m_1}{T}_2^{a_1/2}{s}_1|g_1=1,g_2{=0\rangle }_{a=0}=T_1^{-k_1}{T}_4^{m_1}{T}_2^{a_1/2}{|g_1=1,g_2=0\rangle }_{(0,0,0,2)}\hfill \\ \hfill & =T_1^{-k_1-1}{T}_4^{m_1}{T}_2^{a_1/2}{T}_1{|g_1=1,g_2=0\rangle }_{(0,0,0,2)},\hfill \end{array}$$

(91)

where we used the relations $s_i{T}_i{s}_i=T_{i+1}$, $s_i{T}_j=T_j{s}_i,j\ne i,i+1$ that hold between the $s_i$ transformations and the translation operators [11]. Also, we used that $s_1$ is an identity when acting on $|g_1=1,g_2{=0\rangle }_{(0,0,0,2)}$.

On the right-hand side of Equation (90), we have

$$T_4^{k_3}{T}_2^{-m_3}{\widehat{P}}_1|g_1=1,g_2{=0\rangle }_{a_3}=T_4^{k_3}{T}_2^{-m_3}{T}_1^{-a_3/2}{|g_1=0,g_2=1\rangle }_{(2,0,0,0)},$$

(92)

where we used that ${\widehat{P}}_1{T}_1{\widehat{P}}_1=T_1^{-1}$ and

$$\begin{array}{cc}\hfill {\widehat{P}}_1(g_1=1,g_2=0,g_3=0,g_4=1)& =(g_4=1,g_3=0,g_2=0,g_1=1)=(1,0,0,1),\hfill \\ \hfill {\widehat{P}}_1(a,0,0,2-a)& =(2-a,0,0,a).\hfill \end{array}$$

Comparing the powers of $T_1,T_4,T_2$ in expressions (91) and (92), we obtain three conditions:

$$l_3=1+k_1,m_1=k_3,l_1=m_3,$$

(93)

with $a_i=2l_i,i=1,3$. When these conditions are satisfied, the Relation (86) holds both for the parameters and the solutions provided that

$$T_1|g_1=1,g_2{=0\rangle }_{(0,0,0,2)}={\widehat{P}}_1|g_1=1,g_2{=0\rangle }_{(0,0,0,2)}={|g_1=1,g_2=0\rangle }_{(2,0,0,0)},$$

but since $T_1$ only changes the parameters but not the underlying $F,G$ variables, this relation is an identity and, therefore, the relation (86) also holds. Comparing the Relations (89) and (93), we find a direct relation:

$$l-m-1=m_1,k_1=m+k,m=l_1,$$

between the parameters $k_1,m_1,l_1$ and $k,m,l$. These are the exact values [12] for which it holds that solutions $\pi s_1{T}_2^{-k_1}{T}_4^{m_1}{|F=1,G=0\rangle }_{{\alpha }_{\mathsf{b}}}$ and $T_2^{-k}{T}_4^m{|F=1,G=0\rangle }_{{\alpha }_{\mathsf{a}}}$ are different while the four parameters of the two solutions are equal [12]:

$$\pi s_1\left({\alpha }_{k_1,m_1;\mathsf{b}}\right)={\alpha }_{k,m;\mathsf{a}}.$$

Example 3. 

$${\widehat{P}}_2:(F,G)\to (\mathrm{degenerate}\left(F\right),\mathrm{degenerate}\left(G\right))$$

Let $m=2$ and $k=1$ and $l=3$, then, the corresponding solution is obtained from Relation (58) to be

$$\begin{array}{cc}& T_2^{-1}{T}_4^2{|F=1,G=0\rangle }_{{\alpha }_{\mathsf{a}=6}}=(F={\displaystyle \frac{(x^2+4x+6)(x^3+9x^2+36x+60)}{(x^2+6x+12)(18x+6x^2+x^3+24)}},\hfill \\ & G=2{\displaystyle \frac{(x^2+6x+6)(18x+6x^2+x^3+24)}{(x^2+4x+6)(12x^3+54x^2+96x+72+x^4)}}),\hfill \end{array}$$

with ${\alpha }_i=2(l+k,-k,-m,1-l+m)=2(4,-1,-2,0)=2\pi s_1(l_1+k_1,-k_1,-m_1,1-l_1+m_1)$. The unique solution for $l_1,k_1,m_1$ is: $l_1=-2,k_1=3,m_1=0$. We use the right-hand side of Relation (86) to write the degenerated solution as follows:

$$\begin{array}{cc}\hfill & \pi s_1{T}_2^{-3}{|F=1,G=0\rangle }_{{\alpha }_{\mathsf{a}=-4}}=({\displaystyle \frac{-8x+12+x^2}{-6x+6+x^2}},1),\hfill \\ \hfill & \pi s_{1}2(l_1+k_1,-k_1,-m_1,1-l_1+m_1)=2\pi s_1(1,-3,0,3)=2(4,-1,-2,0).\hfill \end{array}$$

(94)

We will now reproduce the degenerated solution (94) using the ${\widehat{P}}_2$ transposition and the left-hand side of Relation (86). Starting again as above with $m=2,k=1$ and $a=6$ or $l=3$ so that ${\alpha }_i=2(l+k,-k,-m,1-l+m)=2(4,-1,-2,0)$ with $l-1-m=0$ and $F_m^{l-1-m}(-x,2(1+m+k))=F_2^0(-x,2·4)$:

$$\begin{array}{cc}\hfill F_2^0(-x,2·4)& ={\displaystyle \frac{N_0^{\left(1\right)}(x,2(-2))N_{2(3-2)}^{\left(1\right)}(x,-2\left(3\right))}{N_0^{\left(1\right)}(x,-2\left(3\right))N_{2(3-2)}^{\left(1\right)}(x,2(-2))}}={\displaystyle \frac{N_2^{\left(1\right)}(x,-6)}{N_2^{\left(1\right)}(x,-4)}}\hfill \\ & ={\displaystyle \frac{-32x+48+4x^2}{-24x+24+4x^2}}={\displaystyle \frac{-8x+12+x^2}{-6x+6+x^2}}.\hfill \end{array}$$

Since $l-1-m=0$, then $G_m^{l-1-m}(-x,2(1+m+k))=0$, but then the ${\widehat{P}}_2$ transformed solution is ${\widehat{P}}_2\left(G\right)=1-G_m^{l-1-m}(-x,2(1+m+k))$, which is equal to 1.

The above functions $F_2^0(-x,2·4)=(-8x+12+x^2)/(-6x+6+x^2)$ and ${\widehat{P}}_2\left(G\right)=1$ solve the Hamiltonian Equation (2) with ${\alpha }_1=8,{\alpha }_2=-2,{\alpha }_3=-4$ coefficients so that again we obtain acting with the ${\widehat{P}}_2$ automorphism a solution for the same parameters ${\alpha }_i=2(4,-1,-2,0)$ as the original $F_m^k,G_m^k$ solution.

Example 4. 

Here, we illustrate how the action of ${\widehat{P}}_4$ generates degenerated solutions according to a simple prescription:

$${\widehat{P}}_4:(F,G)\to (\mathit{degenerate}\left(F\right),\mathit{degenerate}\left(G\right)).$$

Recall that, as shown in Section 6.2, we have

$${\widehat{P}}_4:{\alpha }_1\leftrightarrow {\alpha }_3\sim l\to -m-k\in {\mathbb{Z}}_{+},m\to -l-k\in {\mathbb{Z}}_{+},$$

and

$$\begin{array}{cc}\hfill {\widehat{P}}_4\left(F_m^k(x,l)\right)& =1-F_{-l-k}^k(-x,-m-k),\hfill \\ \hfill {\widehat{P}}_4\left(G_m^k(x,l)\right)& =G_{-l-k}^k(-x,-m-k),\mathsf{a}=2l.\hfill \end{array}$$

Accordingly, for $m=1,k=1,l=-1$ or $\mathsf{a}=-2$, we obtain

$$\begin{array}{cc}\hfill F_1^1(x,-2)={\displaystyle \frac{{(x-2)}^{2}x}{(x-1)(x^2-4x+2)}}& \stackrel{{\widehat{P}}_4}{⟶}1-F_0^1(-x,-2)=0,\hfill \\ \hfill G_1^1(x,-4)={\displaystyle \frac{(x^2-4x+2)}{(x-2)(x^2-2x+2)}}& \stackrel{{\widehat{P}}_4}{⟶}{G}_0^1(-x,-2)=-{\displaystyle \frac{1}{x+2}}.\hfill \end{array}$$

Both the above results are solutions of the $F,G$ Equation (2) with the same parameters.

Although Equations (86) and (87) follow from each other, they hold for different values of the parameter l, such that $\mathsf{a}=2l$. To show this, we discuss the equality between the parameters shared between the left-hand side of Equation (86) and the parameter of the solution $T_2^k{T}_4^m{|g_1=1,g_2=0\rangle }_a$. The common parameters are as follows:

$$T_2^{-k}{T}_4^m(\mathsf{a},0,0,2-\mathsf{a})={\widehat{P}}_2{T}_2^{-k_2}{T}_4^{m_2}({\mathsf{a}}_2,0,0,2-{\mathsf{a}}_2),$$

or

$$\begin{array}{cc}\hfill (l+k,-k,-m,1-l+m)& ={\widehat{P}}_2(l_2+k_2,-k_2,-m_2,1-l_2+m_2)\hfill \\ & =(l_2+k_2,1-l_2+m_2,-m_2,-k_2),\hfill \end{array}$$

the solution to these conditions is

$$m_2=m,l=k_1+m+1,l_2=k+m_2+1=k+m+1,$$

which implies that both l and $l_2$ are positive.

$$l>m,l_2>m.$$

Repeating the same analysis for the degeneracy connected with ${\widehat{P}}_4$, we start with the equality

$$T_2^{-k}{T}_4^m(\mathsf{a},0,0,2-\mathsf{a})={\widehat{P}}_4{T}_2^{-k_0}{T}_4^{m_0}({\mathsf{a}}_0,0,0,2-{\mathsf{a}}_0),$$

that can be rewritten as

$$\begin{array}{cc}\hfill (l+k,-k,-m,1-l+m)& ={\widehat{P}}_4(l_0+k_0,-k_0,-m_0,1-l_0+m_0)\hfill \\ & =(-m_0,-k_0,l_0+k_0,1-l_0+m_0).\hfill \end{array}$$

This time, it is the k parameter which remains unchanged $k=k_0$ while the expressions for $l,l_0$ become $l=-k-m_0,l_0=-k-m$. Thus, the ${\widehat{P}}_4$ degeneracy occurs when both $l,l_0$ are negative and satisfy $l<-k_0,l<k$.

Thus, to cover all cases of both positive and negative values of the parameter l, one needs to consider the degeneracies induced by both ${\widehat{P}}_2$ and ${\widehat{P}}_4$.

7. The Odd ${\widehat{\mathit{P}}}_{\mathbf{1}}$ and ${\widehat{\mathit{P}}}_{\mathbf{3}}$ Automorphisms

7.1. The Action of ${\widehat{P}}_1$ Automorphism

Consider the two ${\widehat{P}}_1$ transformations

$$G_m^k(x,\mathsf{a})\stackrel{{\widehat{P}}_1}{⟶}{G}_k^m(-x,-a+2)=1-F_m^k(x,\mathsf{a}),$$

(95)

and

$$F_m^k(x,\mathsf{a})\stackrel{{\widehat{P}}_1}{⟶}{F}_k^m(-x,-a+2)=1-G_m^k(x,\mathsf{a}),$$

(96)

where we took into consideration that ${\widehat{P}}_1$ transforms $x\to -x$, $a\to 2-a$ and $m\leftrightarrow k$ (which is equivalent to ${\alpha }_1\leftrightarrow {\alpha }_4$, ${\alpha }_2\leftrightarrow {\alpha }_3$). Simultaneously, we have ${\widehat{P}}_1\left(F\right)=1-G$ and ${\widehat{P}}_1\left(G\right)=1-F$ that we inserted on the right-hand sides.

Using the identities (75) from Section 5.2, we find for $G_k^m(-x,-a+2)$ from (58) using Relation (75):

$$\begin{array}{cc}\hfill G_k^m(-x,-a+2)& =-2m{\displaystyle \frac{R_{k(m+1)}^{\left(m\right)}(x,\mathsf{a})R_{(k+1)(m-1)}^{(m-2)}(x,\mathsf{a}-2)}{R_{mk}^{(m-1)}(x,\mathsf{a})R_{(k+1)m}^{(m-1)}(x,\mathsf{a}-2)}}\hfill \\ \hfill & =1-F_m^k(x,\mathsf{a})=1-{\displaystyle \frac{R_{mk}^{(m-1)}(x,\mathsf{a}-2)R_{m(k+1)}^{(m-1)}(x,\mathsf{a})}{R_{mk}^{(m-1)}(x,\mathsf{a})R_{m(k+1)}^{(m-1)}(x,\mathsf{a}-2)}},\hfill \end{array}$$

(97)

or after multiplying both sides by $R_{km}^{(m-1)}(x,\mathsf{a})/R_{m(k+1)}^{(m-1)}(x,\mathsf{a})$:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{R_{(m-1)(k+1)}^{(m-2)}(x,a-2)R_{(m+1)k}^{\left(m\right)}(x,\mathsf{a})}{R_{m(k+1)}^{(m-1)}(x,\mathsf{a})R_{m(k+1)}^{(m-1)}(x,\mathsf{a}-2)}}\hfill \\ \hfill & ={\displaystyle \frac{1}{2m}}\left({\displaystyle \frac{R_{mk}^{(m-1)}(x,\mathsf{a}-2)}{R_{m(k+1)}^{(m-1)}(x,\mathsf{a}-2)}}-{\displaystyle \frac{R_{km}^{(m-1)}(x,\mathsf{a})}{R_{m(k+1)}^{(m-1)}(x,\mathsf{a})}}\right),\hfill \end{array}$$

(98)

in which we recognize the master recurrence Relation (59)!

Looking back on the remaining relation in Equation (96), it can be shown that it is equivalent to the recurrence Relation (72), the dual one to the master recurrence Relation (59).

This completes the ${\widehat{P}}_1$ symmetry discussion showing how it nicely fits the structure of the solutions we have generated and is fully characterized by the recursion relations. Thus, the ${\widehat{P}}_1$ automorphism is verified as long as the fundamental recursion relations hold and its invariance can be considered a consistency check on the formalism.

7.2. The ${\widehat{P}}_3$ Automorphism

The automorphism ${\widehat{P}}_3$ transforms $x\to -x$ and ${\alpha }_1\leftrightarrow {\alpha }_2$ and ${\alpha }_3\leftrightarrow {\alpha }_4$. For ${\alpha }_i=2(l+k,-k,-m,1-l+m)$, it follows that ${\widehat{P}}_3\left({\alpha }_i\right)=2(-k,l+k,1-l+m,-m)=2(\overline{l}+\overline{k},-\overline{k},-\overline{m},1-\overline{l}+\overline{m})$, which implies

$$k\to \overline{k}=-l-k,m\to \overline{m}=l-1-m.$$

Note that ${\alpha }_1+{\alpha }_2=2l$ and ${\alpha }_3+{\alpha }_4=2(1-l)$ are invariant under ${\widehat{P}}_3$, and so this time, l is invariant! Note that for $\overline{k}$ to be positive. we must have $l<0$ with $\left|l\right|\ge k$. For $\overline{m}$ to be positive, we must have $l>0$ and $l\ge m+1$. These are contradictory requirements and it apears that the ${\widehat{P}}_3$ transformed solution is not obtained by a simply transformation or relabeling of parameters. However, the transformation

$${\widehat{P}}_3:x\to -x,F\to G,G\to F,{\alpha }_1\leftrightarrow {\alpha }_2,{\alpha }_3\leftrightarrow {\alpha }_4,$$

is realized as

$$\begin{array}{cc}& {\widehat{P}}_3{T}_2^{-k}{T}_4^m{|F=1,G=0\rangle }_{(a,0,0,2-a)}=T_2^k{T}_4^{-m}{|F=0,G=1\rangle }_{(0,a,2-a,0)}\hfill \\ & ={\widehat{P}}_3(F_m^k(x,2l),G_m^k(x,2l))=(G_m^k(-x,2l),F_m^k(-x,2l)).\hfill \end{array}$$

Substituting $k\to \overline{k}=-l-k$, $m\to \overline{m}=l-1-m$ with $\overline{l}=l$ into equation

$$\begin{array}{cc}\hfill F_x& =F(F-1)(2G-1)-{\displaystyle \frac{l+k-m}{x}}F+{\displaystyle \frac{l+k}{x}},\hfill \\ \hfill G_x& =-G(G-1)(2F-1)+{\displaystyle \frac{l+k-m-1}{x}}G-{\displaystyle \frac{k}{x}},\hfill \end{array}$$

(99)

followed by the $G\leftrightarrow F$ and $x\to -x$ transformation will leave Equation (99) invariant, verifying the invariance under the ${\widehat{P}}_3$ automorphism.

8. Extending the Method to $\mathit{N}>\mathbf{4}$

Here, we consider the special cases of solutions for higher $N=6$ generated from the specific seed solutions and illustrate the power of the method based on automorphisms to determine the explicit form of the solutions. For convenience, to describe the seed solution in such cases, we will be using the notation based on variables $j_i\left(z\right)$ of the even dressing chain (7).

$N=6$ and the Seed Solutions $j_i\left(z\right)=z/2,i=1,2,3$ and $j_i\left(z\right)={(-1)}^{i+1}z/2,i=4,\dots ,N$

In this subsection, we will consider solutions constructed from seed solutions of the type $j\left(z\right)=(z/2)(1,1,1,-1,1,-1)$ for even $N=6$. Here, we analyze such model for $N=6$ but the construction extends easily to all higher N.

We first consider a basic seed solution $j_i={\displaystyle \frac{z}{2}}(1,1,1,-1,1,-1)$ with the parameters ${\alpha }_i=(2-{\mathsf{a}}_2,{\mathsf{a}}_2,0,0,0,0)$ of the $N=6$ dressing Equation (7). This seed solution solves the higher Painlevé Equation (4) with $g_1=1,g_3=g_5=0,g_2=1,g_4=g_6=0$ and will be denoted by $|g_1=1,g_2{=1\rangle }_{(2-\mathsf{a},\mathsf{a},0,0,0,0)}$. For the above solution, it holds that $j_3+j_4=0,j_4+j_5=0,j_5+j_6=0,j_6+j_1=0$. As discussed in [11], to avoid division by zero, we need to exclude any action by $s_i,s_{i-1}{\pi }^{-1},s_{i+1}\pi$ with $i=3,4,5,6$. For similar reasons, we also need to exclude the translation operators $T_i,T_{i+1}^{-1},i=3,4,5,6$. In conclusion, we are led to the only finite solutions of the form:

$$T_1^{n_1}{T}_2^{n_2}{T}_3^{-n_3}{|g_1=1,g_2=1\rangle }_{(2-\mathsf{a},\mathsf{a},0,0,0,0)},n_1,n_3\in {\mathbb{Z}}_{+},n_2\in \mathbb{Z},$$

with the parameters

$$(2-\mathsf{a}+2n_1-2n_2,\mathsf{a}+2n_2+2n_3,-2n_3,0,0,-2n_1).$$

(100)

One sees that the effect of $T_2$ is only to redefine $\mathsf{a}$ by a shift with $2n_2$ as $T_1$ did in the case of the $N=4$ model. According to the transformation rule (35), we find that the above three translation operators are being connected via the automorphism ${\widehat{P}}_3$:

$${\widehat{P}}_3{T}_3^{-1}{\widehat{P}}_3=T_1,{\widehat{P}}_3{T}_2{\widehat{P}}_3=T_2^{-1}.$$

(101)

We can now investigate the recurrence relations emerging on the orbit of $T_3^{-1}$ to express it in terms of the Kummer polynomials and build the complete solutions utilizing fully the ${\widehat{P}}_3$ automorphism to ensure invariance under: $x\to -x,a\to 2-a,n_2\to n_4,n_4\to n_2$. For the pure $T_1$ and $T_3^{-1}$ orbits of the configuration

$$G_{\mathrm{seed}}^{\left(6\right)}=g_i=(1,1,0,0,0,0),{\alpha }_i^{\left(6\right)}=(2-\mathsf{a},\mathsf{a},0,0,0,0),$$

we find

$$\begin{array}{cc}\hfill \mathbb{T}(k,0;\mathsf{a})& =T_3^{-k}{|G_{\mathrm{seed}}^{\left(6\right)}\rangle }_{{\alpha }_i^{\left(6\right)}}\hfill \\ & =|g_1={\displaystyle \frac{R_k(x,\mathsf{a}-2)}{R_k(x,\mathsf{a})}},g_2=1,g_3=2k{\displaystyle \frac{R_{k-1}(x,\mathsf{a})}{R_k(x,\mathsf{a})}}{,0,0,0\rangle }_{(2-\mathsf{a},\mathsf{a}+2k,-2k,0,0,0)}\hfill \\ \hfill \mathbb{T}(0,m;\mathsf{a})& =T_1^m{|G_{\mathrm{seed}}^{\left(6\right)}\rangle }_{{\alpha }_i^{\left(6\right)}}\hfill \\ & =|g_1=1,g_2={\displaystyle \frac{N_m(x,\mathsf{a}+2)}{N_m(x,\mathsf{a})}},g_3=1,0,0,g_6=-2m{\displaystyle \frac{N_{m-1}(x,\mathsf{a})}{N_m(x,\mathsf{a})}}{\rangle }_{(2-\mathsf{a}+2m,\mathsf{a},0,0,0,-2m)}\hfill \end{array}$$

(102)

We will now generalize the above results and calculate the complete solution

$$\mathbb{T}(k,m;\mathsf{a})=T_3^{-k}{T}_1^m|G_{\mathrm{seed}}^{\left(6\right)}{\rangle }_{{\alpha }_i^{\left(6\right)}}={|g_1^{(k,m)},g_2^{(k,m)},g_3^{(k,m)},0,0,g_6^{(k,m)}\rangle }_{(2-\mathsf{a}+2m,\mathsf{a}+2k,-2k,0,0,-2m)}.$$

First, we observe that on the basis of the result (58) and the pure orbit results for $\mathbb{T}(k,0;\mathsf{a})$ and $\mathbb{T}(0,m;\mathsf{a})$ in (102), it holds that:

$$\begin{array}{cc}\hfill g_2^{(k,m)}(x,\mathsf{a})& ={\displaystyle \frac{R_{mk}^{(m-1)}(x,\mathsf{a}-2)R_{m(k+1)}^{(m-1)}(x,\mathsf{a})}{R_{mk}^{(m-1)}(x,\mathsf{a})R_{m(k+1)}^{(m-1)}(x,\mathsf{a}-2)}},\hfill \\ \hfill g_3^{(k,m)}(x,\mathsf{a})& =2k{\displaystyle \frac{R_{m(k+1)}^{(m-1)}(x,\mathsf{a}-2)R_{(m+1)(k-1)}^{\left(m\right)}(x,\mathsf{a})}{R_{mk}^{(m-1)}(x,\mathsf{a}-2)R_{(m+1)k}^{\left(m\right)}(x,\mathsf{a})}},m,k=0,1,2,\dots .\hfill \end{array}$$

(103)

We will now use the fact that transformations by the $P_3$ reflection automorphism are ${\widehat{P}}_3\left(g_2\right)=g_1$ and ${\widehat{P}}_3\left(g_3\right)=g_6$, together with $x\to -x,a\to 2-a,k\to m,m\to k$, to obtain

$$\begin{array}{cc}\hfill g_1^{(k,m)}(x,\mathsf{a})& ={\displaystyle \frac{R_{mk}^{(m-1)}(x,\mathsf{a})R_{k(m+1)}^{\left(m\right)}(x,\mathsf{a}-2)}{R_{mk}^{(m-1)}(x,\mathsf{a}-2)R_{k(m+1)}^{\left(m\right)}(x,\mathsf{a})}},\hfill \\ \hfill g_6^{(k,m)}(x,\mathsf{a})& =-2m{\displaystyle \frac{R_{k(m+1)}^{\left(m\right)}(x,\mathsf{a})R_{(k+1)(m-1)}^{(m-2)}(x,\mathsf{a}-2)}{R_{mk}^{(m-1)}(x,\mathsf{a})R_{(k+1)m}^{(m-1)}(x,\mathsf{a}-2)}},m,k=0,1,2,\dots .\hfill \end{array}$$

(104)

where in the derivation of the above result, we also used the identity (75).

9. Summary and Outlook

The even-periodicity Painlevé equations exhibit a more extensive symmetry structure compared to the odd-periodicity Painlevé equations. In this work, we introduced additional even and odd reflection automorphisms, each playing distinct roles in determining the rational solutions. Even reflections generate new solutions while preserving the parameters of the Painlevé equations, directly contributing to degeneracy. Conversely, odd reflections act as duality transformations, altering the appearance of the solutions by mapping them into a dual formulation, but maintaining the solutions themselves.

The duality property is crucial for understanding the structure of the solutions and provides a framework for combining the solutions from the simple orbits of a single translation operator into orbits created by the actions of two independent translation operators. Developing this framework to accommodate more than two translation operators will be essential for a complete characterization of more complex solutions, including those involving the Umemura polynomials.

In the future, we plan to provide proofs for the $N>4$ models to establish a general framework for generating degeneracy by even reflections and to provide further details on constructing all rational solutions for higher values of N.

Additionally, it will be important to establish a theory of discrete Painlevé structures emerging from the kind of translation operators explored in this paper.