# §32.8 Rational Solutions

## §32.8(i) Introduction

$\mbox{P}_{\mbox{\scriptsize II}}$$\mbox{P}_{\mbox{\scriptsize VI}}$ possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants. See Airault (1979).

## §32.8(ii) Second Painlevé Equation

Rational solutions of $\mbox{P}_{\mbox{\scriptsize II}}$ exist for $\alpha=n(\in\mathbb{Z})$ and are generated using the seed solution $w(z;0)=0$ and the Bäcklund transformations (32.7.1) and (32.7.2). The first four are

 32.8.1 $w(z;1)=-\ifrac{1}{z},$ ⓘ Symbols: $z$: real Permalink: http://dlmf.nist.gov/32.8.E1 Encodings: TeX, pMML, png See also: Annotations for §32.8(ii), §32.8 and Ch.32
 32.8.2 $w(z;2)=\frac{1}{z}-\frac{3z^{2}}{z^{3}+4},$ ⓘ Symbols: $z$: real Permalink: http://dlmf.nist.gov/32.8.E2 Encodings: TeX, pMML, png See also: Annotations for §32.8(ii), §32.8 and Ch.32
 32.8.3 $w(z;3)=\frac{3z^{2}}{z^{3}+4}-\frac{6z^{2}(z^{3}+10)}{z^{6}+20z^{3}-80},$ ⓘ Symbols: $z$: real Permalink: http://dlmf.nist.gov/32.8.E3 Encodings: TeX, pMML, png See also: Annotations for §32.8(ii), §32.8 and Ch.32
 32.8.4 $w(z;4)=-\frac{1}{z}+\frac{6z^{2}(z^{3}+10)}{z^{6}+20z^{3}-80}-\frac{9z^{5}(z^{% 3}+40)}{z^{9}+60z^{6}+11200}.$ ⓘ Symbols: $z$: real Permalink: http://dlmf.nist.gov/32.8.E4 Encodings: TeX, pMML, png See also: Annotations for §32.8(ii), §32.8 and Ch.32

More generally,

 32.8.5 $w(z;n)=\frac{\mathrm{d}}{\mathrm{d}z}\left(\ln\left(\frac{Q_{n-1}(z)}{Q_{n}(z)% }\right)\right),$

where the $Q_{n}(z)$ are monic polynomials (coefficient of highest power of $z$ is $1$) satisfying

 32.8.6 $Q_{n+1}(z)Q_{n-1}(z)={zQ_{n}^{2}(z)+4\left(Q_{n}^{\prime}(z)\right)^{2}-4Q_{n}% (z)Q_{n}^{\prime\prime}(z)},$ ⓘ Symbols: $n$: integer, $z$: real and $Q_{n}(z)$: polynomial Permalink: http://dlmf.nist.gov/32.8.E6 Encodings: TeX, pMML, png See also: Annotations for §32.8(ii), §32.8 and Ch.32

with $Q_{0}(z)=1$, $Q_{1}(z)=z$. Thus

 32.8.7 $\displaystyle Q_{2}(z)$ $\displaystyle=z^{3}+4,$ $\displaystyle Q_{3}(z)$ $\displaystyle=z^{6}+20z^{3}-80,$ $\displaystyle Q_{4}(z)$ $\displaystyle=z^{10}+60z^{7}+11200z,$ $\displaystyle Q_{5}(z)$ $\displaystyle=z^{15}+140z^{12}+2800z^{9}+78400z^{6}-3\;13600z^{3}-62\;72000,$ $\displaystyle Q_{6}(z)$ $\displaystyle=z^{21}+280z^{18}+18480z^{15}+6\;27200z^{12}-172\;48000z^{9}+1448% 8\;32000z^{6}+1\;93177\;60000z^{3}-3\;86355\;20000.$ ⓘ Symbols: $z$: real and $Q_{n}(z)$: polynomial Permalink: http://dlmf.nist.gov/32.8.E7 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §32.8(ii), §32.8 and Ch.32

Next, let $p_{m}(z)$ be the polynomials defined by $p_{m}(z)=0$ for $m<0$, and

 32.8.8 $\sum_{m=0}^{\infty}p_{m}(z)\lambda^{m}=\exp\left(z\lambda-\tfrac{4}{3}\lambda^% {3}\right).$ ⓘ Symbols: $\exp\NVar{z}$: exponential function, $m$: integer, $z$: real and $p_{m}(z)$: polynomials Permalink: http://dlmf.nist.gov/32.8.E8 Encodings: TeX, pMML, png See also: Annotations for §32.8(ii), §32.8 and Ch.32

Then for $n\geq 2$

 32.8.9 $w(z;n)=\frac{\mathrm{d}}{\mathrm{d}z}\left(\ln\left(\frac{\tau_{n-1}(z)}{\tau_% {n}(z)}\right)\right),$

where $\tau_{n}(z)$ is the $n\times n$ determinant

 32.8.10 $\tau_{n}(z)=\begin{vmatrix}p_{1}(z)&p_{3}(z)&\cdots&p_{2n-1}(z)\\ p_{1}^{\prime}(z)&p_{3}^{\prime}(z)&\cdots&p_{2n-1}^{\prime}(z)\\ \vdots&\vdots&\ddots&\vdots\\ p_{1}^{(n-1)}(z)&p_{3}^{(n-1)}(z)&\cdots&p_{2n-1}^{(n-1)}(z)\end{vmatrix}.$ ⓘ Symbols: $\det$: determinant, $n$: integer, $z$: real, $p_{m}(z)$: polynomials and $\tau_{n}(z)$: determinant Permalink: http://dlmf.nist.gov/32.8.E10 Encodings: TeX, pMML, png See also: Annotations for §32.8(ii), §32.8 and Ch.32

For plots of the zeros of $Q_{n}(z)$ see Clarkson and Mansfield (2003).

## §32.8(iii) Third Painlevé Equation

Special rational solutions of $\mbox{P}_{\mbox{\scriptsize III}}$ are

 32.8.11 $w(z;\mu,-\mu\kappa^{2},\lambda,-\lambda\kappa^{4})=\kappa,$ ⓘ Symbols: $z$: real, $\kappa$: contstant, $\lambda$: contstant and $\mu$: contstant Permalink: http://dlmf.nist.gov/32.8.E11 Encodings: TeX, pMML, png See also: Annotations for §32.8(iii), §32.8 and Ch.32
 32.8.12 $w(z;0,-\mu,0,\mu\kappa)=\kappa z,$ ⓘ Symbols: $z$: real, $\kappa$: contstant and $\mu$: contstant Permalink: http://dlmf.nist.gov/32.8.E12 Encodings: TeX, pMML, png See also: Annotations for §32.8(iii), §32.8 and Ch.32
 32.8.13 $w(z;2\kappa+3,-2\kappa+1,1,-1)=\dfrac{z+\kappa}{z+\kappa+1},$ ⓘ Symbols: $z$: real and $\kappa$: contstant Permalink: http://dlmf.nist.gov/32.8.E13 Encodings: TeX, pMML, png See also: Annotations for §32.8(iii), §32.8 and Ch.32

with $\kappa$, $\lambda$, and $\mu$ arbitrary constants.

In the general case assume $\gamma\delta\neq 0$, so that as in §32.2(ii) we may set $\gamma=1$ and $\delta=-1$. Then $\mbox{P}_{\mbox{\scriptsize III}}$ has rational solutions iff

 32.8.14 $\alpha\pm\beta=4n,$ ⓘ Symbols: $n$: integer, $\alpha$: arbitrary constant and $\beta$: arbitrary constant Permalink: http://dlmf.nist.gov/32.8.E14 Encodings: TeX, pMML, png See also: Annotations for §32.8(iii), §32.8 and Ch.32

with $n\in\mathbb{Z}$. These solutions have the form

 32.8.15 $w(z)=\ifrac{P_{m}(z)}{Q_{m}(z)},$ ⓘ Symbols: $m$: integer, $z$: real, $P_{m}(z)$: polynomials and $Q_{m}(z)$: polynomials Permalink: http://dlmf.nist.gov/32.8.E15 Encodings: TeX, pMML, png See also: Annotations for §32.8(iii), §32.8 and Ch.32

where $P_{m}(z)$ and $Q_{m}(z)$ are polynomials of degree $m$, with no common zeros.

For examples and plots see Milne et al. (1997); also Clarkson (2003a). For determinantal representations see Kajiwara and Masuda (1999).

## §32.8(iv) Fourth Painlevé Equation

Special rational solutions of $\mbox{P}_{\mbox{\scriptsize IV}}$ are

 32.8.16 $\displaystyle w_{1}(z;\pm 2,-2)$ $\displaystyle=\pm\ifrac{1}{z},$ ⓘ Symbols: $z$: real Permalink: http://dlmf.nist.gov/32.8.E16 Encodings: TeX, pMML, png See also: Annotations for §32.8(iv), §32.8 and Ch.32 32.8.17 $\displaystyle w_{2}(z;0,-2)$ $\displaystyle=-2z,$ ⓘ Symbols: $z$: real Permalink: http://dlmf.nist.gov/32.8.E17 Encodings: TeX, pMML, png See also: Annotations for §32.8(iv), §32.8 and Ch.32 32.8.18 $\displaystyle w_{3}(z;0,-\tfrac{2}{9})$ $\displaystyle=-\tfrac{2}{3}z.$ ⓘ Symbols: $z$: real Permalink: http://dlmf.nist.gov/32.8.E18 Encodings: TeX, pMML, png See also: Annotations for §32.8(iv), §32.8 and Ch.32

There are also three families of solutions of $\mbox{P}_{\mbox{\scriptsize IV}}$ of the form

 32.8.19 $w_{1}(z;\alpha_{1},\beta_{1})=\ifrac{P_{1,n-1}(z)}{Q_{1,n}(z)},$
 32.8.20 $w_{2}(z;\alpha_{2},\beta_{2})=-2z+(\ifrac{P_{2,n-1}(z)}{Q_{2,n}(z)}),$
 32.8.21 $w_{3}(z;\alpha_{3},\beta_{3})=-\tfrac{2}{3}z+(\ifrac{P_{3,n-1}(z)}{Q_{3,n}(z)}),$

where $P_{j,n-1}(z)$ and $Q_{j,n}(z)$ are polynomials of degrees $n-1$ and $n$, respectively, with no common zeros.

In general, $\mbox{P}_{\mbox{\scriptsize IV}}$ has rational solutions iff either

 32.8.22 $\displaystyle\alpha$ $\displaystyle=m,$ $\displaystyle\beta$ $\displaystyle=-2(1+2n-m)^{2},$ ⓘ Symbols: $m$: integer, $n$: integer, $\alpha$: arbitrary constant and $\beta$: arbitrary constant Referenced by: §32.8(iv) Permalink: http://dlmf.nist.gov/32.8.E22 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §32.8(iv), §32.8 and Ch.32

or

 32.8.23 $\displaystyle\mspace{12.0mu}\alpha$ $\displaystyle=m,$ $\displaystyle\beta$ $\displaystyle=-2(\tfrac{1}{3}+2n-m)^{2},$ ⓘ Symbols: $m$: integer, $n$: integer, $\alpha$: arbitrary constant and $\beta$: arbitrary constant Permalink: http://dlmf.nist.gov/32.8.E23 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §32.8(iv), §32.8 and Ch.32

with $m,n\in\mathbb{Z}$. The rational solutions when the parameters satisfy (32.8.22) are special cases of §32.10(iv).

For examples and plots see Bassom et al. (1995); also Clarkson (2003b). For determinantal representations see Kajiwara and Ohta (1998) and Noumi and Yamada (1999).

## §32.8(v) Fifth Painlevé Equation

Special rational solutions of $\mbox{P}_{\mbox{\scriptsize V}}$ are

 32.8.24 $w(z;\tfrac{1}{2},-\tfrac{1}{2}\mu^{2},\kappa(2-\mu),-\tfrac{1}{2}\kappa^{2})=% \kappa z+\mu,$ ⓘ Symbols: $z$: real, $\kappa$: constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.8.E24 Encodings: TeX, pMML, png See also: Annotations for §32.8(v), §32.8 and Ch.32
 32.8.25 $w(z;\tfrac{1}{2},\kappa^{2}\mu,2\kappa\mu,\mu)=\kappa/(z+\kappa),$ ⓘ Symbols: $z$: real, $\kappa$: constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.8.E25 Encodings: TeX, pMML, png See also: Annotations for §32.8(v), §32.8 and Ch.32
 32.8.26 $w(z;\tfrac{1}{8},-\tfrac{1}{8},-\kappa\mu,\mu)=(\kappa+z)/(\kappa-z),$ ⓘ Symbols: $z$: real, $\kappa$: constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.8.E26 Encodings: TeX, pMML, png See also: Annotations for §32.8(v), §32.8 and Ch.32

with $\kappa$ and $\mu$ arbitrary constants.

In the general case assume $\delta\neq 0$, so that as in §32.2(ii) we may set $\delta=-\tfrac{1}{2}$. Then $\mbox{P}_{\mbox{\scriptsize V}}$ has a rational solution iff one of the following holds with $m,n\in\mathbb{Z}$ and $\varepsilon=\pm 1$:

1. (a)

$\alpha=\tfrac{1}{2}(m+\varepsilon\gamma)^{2}$ and $\beta=-\tfrac{1}{2}n^{2}$, where $n>0$, $m+n$ is odd, and $\alpha\neq 0$ when $|m|.

2. (b)

$\alpha=\tfrac{1}{2}n^{2}$ and $\beta=-\tfrac{1}{2}(m+\varepsilon\gamma)^{2}$, where $n>0$, $m+n$ is odd, and $\beta\neq 0$ when $|m|.

3. (c)

$\alpha=\tfrac{1}{2}a^{2}$, $\beta=-\tfrac{1}{2}(a+n)^{2}$, and $\gamma=m$, with $m+n$ even.

4. (d)

$\alpha=\tfrac{1}{2}(b+n)^{2}$, $\beta=-\tfrac{1}{2}b^{2}$, and $\gamma=m$, with $m+n$ even.

5. (e)

$\alpha=\tfrac{1}{8}(2m+1)^{2}$, $\beta=-\tfrac{1}{8}(2n+1)^{2}$, and $\gamma\notin\mathbb{Z}$.

These rational solutions have the form

 32.8.27 $w(z)=\lambda z+\mu+(\ifrac{P_{n-1}(z)}{Q_{n}(z)}),$ ⓘ Symbols: $n$: integer, $z$: real, $\mu$: constant, $P_{n}(z)$: polynomials and $Q_{n}(z)$: polynomials Permalink: http://dlmf.nist.gov/32.8.E27 Encodings: TeX, pMML, png See also: Annotations for §32.8(v), §32.8 and Ch.32

where $\lambda$, $\mu$ are constants, and $P_{n-1}(z)$, $Q_{n}(z)$ are polynomials of degrees $n-1$ and $n$, respectively, with no common zeros. Cases (a) and (b) are special cases of §32.10(v).

For examples and plots see Clarkson (2005). For determinantal representations see Masuda et al. (2002). For the case $\delta=0$ see Airault (1979) and Lukaševič (1968).

## §32.8(vi) Sixth Painlevé Equation

Special rational solutions of $\mbox{P}_{\mbox{\scriptsize VI}}$ are

 32.8.28 $w(z;\mu,-\mu\kappa^{2},\tfrac{1}{2},\tfrac{1}{2}-\mu(\kappa-1)^{2})=\kappa z,$ ⓘ Symbols: $z$: real, $\kappa$: constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.8.E28 Encodings: TeX, pMML, png See also: Annotations for §32.8(vi), §32.8 and Ch.32
 32.8.29 $\displaystyle w(z;0,0,2,0)$ $\displaystyle=\kappa z^{2},$ ⓘ Symbols: $z$: real and $\kappa$: constant Permalink: http://dlmf.nist.gov/32.8.E29 Encodings: TeX, pMML, png See also: Annotations for §32.8(vi), §32.8 and Ch.32 32.8.30 $\displaystyle w(z;0,0,\tfrac{1}{2},-\tfrac{3}{2})$ $\displaystyle=\ifrac{\kappa}{z},$ ⓘ Symbols: $z$: real and $\kappa$: constant Permalink: http://dlmf.nist.gov/32.8.E30 Encodings: TeX, pMML, png See also: Annotations for §32.8(vi), §32.8 and Ch.32 32.8.31 $\displaystyle w(z;0,0,2,-4)$ $\displaystyle=\ifrac{\kappa}{z^{2}},$ ⓘ Symbols: $z$: real and $\kappa$: constant Permalink: http://dlmf.nist.gov/32.8.E31 Encodings: TeX, pMML, png See also: Annotations for §32.8(vi), §32.8 and Ch.32
 32.8.32 $w(z;\tfrac{1}{2}(\kappa+\mu)^{2},-\tfrac{1}{2},\tfrac{1}{2}(\mu-1)^{2},\tfrac{% 1}{2}\kappa(2-\kappa))=\dfrac{z}{\kappa+\mu z},$ ⓘ Symbols: $z$: real, $\kappa$: constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.8.E32 Encodings: TeX, pMML, png See also: Annotations for §32.8(vi), §32.8 and Ch.32

with $\kappa$ and $\mu$ arbitrary constants.

In the general case, $\mbox{P}_{\mbox{\scriptsize VI}}$ has rational solutions if

 32.8.33 $a+b+c+d=2n+1,$ ⓘ Symbols: $n$: integer Permalink: http://dlmf.nist.gov/32.8.E33 Encodings: TeX, pMML, png See also: Annotations for §32.8(vi), §32.8 and Ch.32

where $n\in\mathbb{Z}$, $a=\varepsilon_{1}\sqrt{2\alpha}$, $b=\varepsilon_{2}\sqrt{-2\beta}$, $c=\varepsilon_{3}\sqrt{2\gamma}$, and $d=\varepsilon_{4}\sqrt{1-2\delta}$, with $\varepsilon_{j}=\pm 1$, $j=1,2,3,4$, independently, and at least one of $a$, $b$, $c$ or $d$ is an integer. These are special cases of §32.10(vi).