§32.7 Bäcklund Transformations

Keywords:: Bäcklund transformations, Painlevé equations, Painlevé transcendents
Permalink:: http://dlmf.nist.gov/32.7

§32.7(i) Definition
§32.7(ii) Second Painlevé Equation
§32.7(iii) Third Painlevé Equation
§32.7(iv) Fourth Painlevé Equation
§32.7(v) Fifth Painlevé Equation
§32.7(vi) Relationship Between the Third and Fifth Painlevé Equations
§32.7(vii) Sixth Painlevé Equation
§32.7(viii) Affine Weyl Groups

§32.7(i) Definition

Notes:: See Fokas and Ablowitz (1982).
Permalink:: http://dlmf.nist.gov/32.7.i

With the exception of $\mbox{P}_{{\mbox{\scriptsize I}}}$ , a Bäcklund transformation relates a Painlevé transcendent of one type either to another of the same type but with different values of the parameters, or to another type.

§32.7(ii) Second Painlevé Equation

Notes:: See Gambier (1910), Lukaševič (1971).
Permalink:: http://dlmf.nist.gov/32.7.ii

Let $w=w(z;\alpha)$ be a solution of $\mbox{P}_{{\mbox{\scriptsize II}}}$ . Then the transformations

32.7.1 $\mathcal{S}:\enskip w(z;-\alpha)=-w,$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant and $\mathcal{S}$ : transformation
Referenced by:: §32.10(ii), §32.8(ii)
Permalink:: http://dlmf.nist.gov/32.7.E1
Encodings:: TeX, pMML, png

and

32.7.2 $\mathcal{T}^{{\pm}}:\enskip w(z;\alpha\pm 1)=-w-\frac{2\alpha\pm 1}{2w^{2}\pm 2w^{{\prime}}+z},$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant and $\mathcal{T}^{{\pm}}$ : transformation
Referenced by:: §32.10(ii), §32.8(ii)
Permalink:: http://dlmf.nist.gov/32.7.E2
Encodings:: TeX, pMML, png

furnish solutions of $\mbox{P}_{{\mbox{\scriptsize II}}}$ , provided that $\alpha\neq\mp\tfrac{1}{2}$ . $\mbox{P}_{{\mbox{\scriptsize II}}}$ also has the special transformation

32.7.3 $W(\zeta;\tfrac{1}{2}\varepsilon)=\frac{2^{{-1/3}}\varepsilon}{w(z;0)}\frac{d}{dz}w(z;0),$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : real, $W(\zeta,\varepsilon/2)$ : transformation and $\varepsilon=\pm 1$
Permalink:: http://dlmf.nist.gov/32.7.E3
Encodings:: TeX, pMML, png

or equivalently,

32.7.4 $w^{2}(z;0)=2^{{-1/3}}\left(W^{2}(\zeta;\tfrac{1}{2}\varepsilon)-\varepsilon\frac{d}{d\zeta}W(\zeta;\tfrac{1}{2}\varepsilon)+\tfrac{1}{2}\zeta\right),$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : real, $W(\zeta,\varepsilon/2)$ : transformation and $\varepsilon=\pm 1$
Permalink:: http://dlmf.nist.gov/32.7.E4
Encodings:: TeX, pMML, png

with $\zeta=-2^{{1/3}}z$ and $\varepsilon=\pm 1$ , where $W(\zeta;\tfrac{1}{2}\varepsilon)$ satisfies $\mbox{P}_{{\mbox{\scriptsize II}}}$ with $z=\zeta$ , $\alpha=\tfrac{1}{2}\varepsilon$ , and $w(z;0)$ satisfies $\mbox{P}_{{\mbox{\scriptsize II}}}$ with $\alpha=0$ .

The solutions $w_{{\alpha}}=w(z;\alpha)$ , $w_{{\alpha\pm 1}}=w(z;\alpha\pm 1)$ , satisfy the nonlinear recurrence relation

32.7.5 $\frac{\alpha+\tfrac{1}{2}}{w_{{\alpha+1}}+w_{{\alpha}}}+\frac{\alpha-\tfrac{1}{2}}{w_{{\alpha}}+w_{{\alpha-1}}}+2w_{{\alpha}}^{2}+z=0.$

Symbols:: $z$ : real and $\alpha$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.7.E5
Encodings:: TeX, pMML, png

See Fokas et al. (1993).

§32.7(iii) Third Painlevé Equation

Notes:: See Fokas and Ablowitz (1982), Gromak (1975); also Gromak et al. (2002, §34).
Permalink:: http://dlmf.nist.gov/32.7.iii

Let $w_{j}=w(z;\alpha _{j},\beta _{j},\gamma _{j},\delta _{j})$ , $j=0,1,2$ , be solutions of $\mbox{P}_{{\mbox{\scriptsize III}}}$ with

32.7.6 $(\alpha _{1},\beta _{1},\gamma _{1},\delta _{1})=(-\alpha _{0},-\beta _{0},\gamma _{0},\delta _{0}),$

Symbols:: $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.7.E6
Encodings:: TeX, pMML, png

32.7.7 $(\alpha _{2},\beta _{2},\gamma _{2},\delta _{2})=(-\beta _{0},-\alpha _{0},-\delta _{0},-\gamma _{0}).$

Symbols:: $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.7.E7
Encodings:: TeX, pMML, png

Then

32.7.8 $\mathcal{S}_{1}:\enskip w_{1}=-w_{0},$

Symbols:: $\mathcal{S}_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E8
Encodings:: TeX, pMML, png

32.7.9 $\mathcal{S}_{2}:\enskip w_{2}=\ifrac{1}{w_{0}}.$

Symbols:: $\mathcal{S}_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E9
Encodings:: TeX, pMML, png

Next, let $W_{j}=W(z;\alpha _{j},\beta _{j},1,-1)$ , $j=0,1,2,3,4$ , be solutions of $\mbox{P}_{{\mbox{\scriptsize III}}}$ with

32.7.10

$\alpha _{1}=\alpha _{3}=\alpha _{0}+2,$

$\alpha _{2}=\alpha _{4}=\alpha _{0}-2,$

$\beta _{1}=\beta _{2}=\beta _{0}+2,$

$\beta _{3}=\beta _{4}=\beta _{0}-2.$

Symbols:: $\alpha$ : arbitrary constant and $\beta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.7.E10
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

Then

32.7.11 $\mathcal{T}_{1}:\enskip W_{1}=\frac{zW_{0}^{{\prime}}+zW_{0}^{2}-\beta W_{0}-W_{0}+z}{W_{0}(zW_{0}^{{\prime}}+zW_{0}^{2}+\alpha W_{0}+W_{0}+z)},$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $W_{j}$ : transformation and $\mathcal{T}_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E11
Encodings:: TeX, pMML, png

32.7.12 $\mathcal{T}_{2}:\enskip W_{2}=-\frac{zW_{0}^{{\prime}}-zW_{0}^{2}-\beta W_{0}-W_{0}+z}{W_{0}(zW_{0}^{{\prime}}-zW_{0}^{2}-\alpha W_{0}+W_{0}+z)},$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $W_{j}$ : transformation and $\mathcal{T}_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E12
Encodings:: TeX, pMML, png

32.7.13 $\mathcal{T}_{3}:\enskip W_{3}=-\frac{zW_{0}^{{\prime}}+zW_{0}^{2}+\beta W_{0}-W_{0}-z}{W_{0}(zW_{0}^{{\prime}}+zW_{0}^{2}+\alpha W_{0}+W_{0}-z)},$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $W_{j}$ : transformation and $\mathcal{T}_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E13
Encodings:: TeX, pMML, png

32.7.14 $\mathcal{T}_{4}:\enskip W_{4}=\frac{zW_{0}^{{\prime}}-zW_{0}^{2}+\beta W_{0}-W_{0}-z}{W_{0}(zW_{0}^{{\prime}}-zW_{0}^{2}-\alpha W_{0}+W_{0}-z)}.$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $W_{j}$ : transformation and $\mathcal{T}_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E14
Encodings:: TeX, pMML, png

See Milne et al. (1997).

If $\gamma=0$ and $\alpha\delta\neq 0$ , then set $\alpha=1$ and $\delta=-1$ , without loss of generality. Let $u_{j}=w(z;1,\beta _{j},0,-1)$ , $j=0,5,6$ , be solutions of $\mbox{P}_{{\mbox{\scriptsize III}}}$ with

32.7.15

$\beta _{5}=\beta _{0}+2,$

$\beta _{6}=\beta _{0}-2.$

Symbols:: $\beta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.7.E15
Encodings:: TeX, TeX, pMML, pMML, png, png

Then

32.7.16 $\mathcal{T}_{5}:\enskip u_{5}=\ifrac{(zu_{0}^{{\prime}}+z-(\beta _{0}+1)u_{0})}{u_{0}^{2}},$

Symbols:: $z$ : real, $\beta$ : arbitrary constant, $\mathcal{T}_{j}$ : transformation and $u_{j}$
Permalink:: http://dlmf.nist.gov/32.7.E16
Encodings:: TeX, pMML, png

32.7.17 $\mathcal{T}_{6}:\enskip u_{6}=-\ifrac{(zu_{0}^{{\prime}}-z+(\beta _{0}-1)u_{0})}{u_{0}^{2}}.$

Symbols:: $z$ : real, $\beta$ : arbitrary constant, $\mathcal{T}_{j}$ : transformation and $u_{j}$
Permalink:: http://dlmf.nist.gov/32.7.E17
Encodings:: TeX, pMML, png

Similar results hold for $\mbox{P}_{{\mbox{\scriptsize III}}}$ with $\delta=0$ and $\beta\gamma\neq 0$ .

Furthermore,

32.7.18

$w(z;a,b,0,0)=W^{2}(\zeta;0,0,a,b),$

$z=\tfrac{1}{2}\zeta^{2}.$

Symbols:: $z$ : real and $W(\zeta,\varepsilon/2)$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E18
Encodings:: TeX, TeX, pMML, pMML, png, png

§32.7(iv) Fourth Painlevé Equation

Notes:: See Lukaševič (1967a), Gromak (1978, 1987); also Gromak et al. (2002, §25).
Permalink:: http://dlmf.nist.gov/32.7.iv

Let $w_{0}=w(z;\alpha _{0},\beta _{0})$ and $w_{j}^{{\pm}}=w(z;\alpha _{j}^{{\pm}},\beta _{j}^{{\pm}})$ , $j=1,2,3,4$ , be solutions of $\mbox{P}_{{\mbox{\scriptsize IV}}}$ with

32.7.19

$\alpha _{1}^{{\pm}}=\tfrac{1}{4}\left(2-2\alpha _{0}\pm 3\sqrt{-2\beta _{0}}\right),$

$\beta _{1}^{{\pm}}=-\tfrac{1}{2}\left(1+\alpha _{0}\pm\tfrac{1}{2}\sqrt{-2\beta _{0}}\right)^{2},$

$\alpha _{2}^{{\pm}}=-\tfrac{1}{4}\left(2+2\alpha _{0}\pm 3\sqrt{-2\beta _{0}}\right),$

$\beta _{2}^{{\pm}}=-\tfrac{1}{2}\left(1-\alpha _{0}\pm\tfrac{1}{2}\sqrt{-2\beta _{0}}\right)^{2},$

$\alpha _{3}^{{\pm}}=\tfrac{3}{2}-\tfrac{1}{2}\alpha _{0}\mp\tfrac{3}{4}\sqrt{-2\beta _{0}},$

$\beta _{3}^{{\pm}}=-\tfrac{1}{2}\left(1-\alpha _{0}\pm\tfrac{1}{2}\sqrt{-2\beta _{0}}\right)^{2},$

$\alpha _{4}^{{\pm}}=-\tfrac{3}{2}-\tfrac{1}{2}\alpha _{0}\mp\tfrac{3}{4}\sqrt{-2\beta _{0}},$

$\beta _{4}^{{\pm}}=-\tfrac{1}{2}\left(-1-\alpha _{0}\pm\tfrac{1}{2}\sqrt{-2\beta _{0}}\right)^{2}.$

Symbols:: $\alpha$ : arbitrary constant and $\beta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.7.E19
Encodings:: TeX, TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png, png

Then

32.7.20 $\mathcal{T}_{1}^{{\pm}}:\enskip w_{1}^{{\pm}}=\frac{w_{0}^{{\prime}}-w_{0}^{2}-2zw_{0}\mp\sqrt{-2\beta _{0}}}{2w_{0}},$

Symbols:: $z$ : real, $\beta$ : arbitrary constant and $\mathcal{T}_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E20
Encodings:: TeX, pMML, png

32.7.21 $\mathcal{T}_{2}^{{\pm}}:\enskip w_{2}^{{\pm}}=-\frac{w_{0}^{{\prime}}+w_{0}^{2}+2zw_{0}\mp\sqrt{-2\beta _{0}}}{2w_{0}},$

Symbols:: $z$ : real, $\beta$ : arbitrary constant and $\mathcal{T}_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E21
Encodings:: TeX, pMML, png

32.7.22 $\mathcal{T}_{3}^{{\pm}}:\enskip w_{3}^{{\pm}}=w_{0}+\frac{2\left(1-\alpha _{0}\mp\tfrac{1}{2}\sqrt{-2\beta _{0}}\right)w_{0}}{w_{0}^{{\prime}}\pm\sqrt{-2\beta _{0}}+2zw_{0}+w_{0}^{2}},$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant and $\mathcal{T}_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E22
Encodings:: TeX, pMML, png

32.7.23 $\mathcal{T}_{4}^{{\pm}}:\enskip w_{4}^{{\pm}}=w_{0}+\frac{2\left(1+\alpha _{0}\pm\tfrac{1}{2}\sqrt{-2\beta _{0}}\right)w_{0}}{w_{0}^{{\prime}}\mp\sqrt{-2\beta _{0}}-2zw_{0}-w_{0}^{2}},$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant and $\mathcal{T}_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E23
Encodings:: TeX, pMML, png

valid when the denominators are nonzero, and where the upper signs or the lower signs are taken throughout each transformation. See Bassom et al. (1995).

§32.7(v) Fifth Painlevé Equation

Notes:: See Gromak (1976); also Gromak et al. (2002, §39).
Permalink:: http://dlmf.nist.gov/32.7.v

Let $w_{j}(z_{j})=w(z_{j};\alpha _{j},\beta _{j},\gamma _{j},\delta _{j})$ , $j=0,1,2$ , be solutions of $\mbox{P}_{{\mbox{\scriptsize V}}}$ with

32.7.24

$z_{1}=-z_{0},$

$z_{2}=z_{0},$

$(\alpha _{1},\beta _{1},\gamma _{1},\delta _{1})=(\alpha _{0},\beta _{0},-\gamma _{0},\delta _{0}),$

$(\alpha _{2},\beta _{2},\gamma _{2},\delta _{2})=(-\beta _{0},-\alpha _{0},-\gamma _{0},\delta _{0}).$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.7.E24
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

Then

32.7.25 $\mathcal{S}_{1}:\enskip w_{1}(z_{1})=w(z_{0}),$

Symbols:: $z$ : real and $\mathcal{S}_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E25
Encodings:: TeX, pMML, png

32.7.26 $\mathcal{S}_{2}:\enskip w_{2}(z_{2})=\ifrac{1}{w(z_{0})}.$

Symbols:: $z$ : real and $\mathcal{S}_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E26
Encodings:: TeX, pMML, png

Let $W_{0}=W(z;\alpha _{0},\beta _{0},\gamma _{0},-\tfrac{1}{2})$ and $W_{1}=W(z;\alpha _{1},\beta _{1},\gamma _{1},-\tfrac{1}{2})$ be solutions of $\mbox{P}_{{\mbox{\scriptsize V}}}$ , where

32.7.27

$\alpha _{1}=\tfrac{1}{8}\left(\gamma _{0}+\varepsilon _{1}\left(1-\varepsilon _{3}\sqrt{-2\beta _{0}}-\varepsilon _{2}\sqrt{2\alpha _{0}}\right)\right)^{2},$

$\beta _{1}=-\tfrac{1}{8}\left(\gamma _{0}-\varepsilon _{1}\left(1-\varepsilon _{3}\sqrt{-2\beta _{0}}-\varepsilon _{2}\sqrt{2\alpha _{0}}\right)\right)^{2},$

$\gamma _{1}=\varepsilon _{1}\left(\varepsilon _{3}\sqrt{-2\beta _{0}}-\varepsilon _{2}\sqrt{2\alpha _{0}}\right),$

Symbols:: $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\varepsilon _{j}=\pm 1$
Permalink:: http://dlmf.nist.gov/32.7.E27
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

and $\varepsilon _{j}=\pm 1$ , $j=1,2,3$ , independently. Also let

32.7.28 $\Phi=zW_{0}^{{\prime}}-\varepsilon _{2}\sqrt{2\alpha _{0}}W_{0}^{2}+\varepsilon _{3}\sqrt{-2\beta _{0}}+\left(\varepsilon _{2}\sqrt{2\alpha _{0}}-\varepsilon _{3}\sqrt{-2\beta _{0}}+\varepsilon _{1}z\right)W_{0},$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\varepsilon _{j}=\pm 1$ , $\Phi$ and $W_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E28
Encodings:: TeX, pMML, png

and assume $\Phi\neq 0$ . Then

32.7.29 $\mathcal{T}_{{\varepsilon _{1},\varepsilon _{2},\varepsilon _{3}}}:\enskip W_{1}=\ifrac{(\Phi-2\varepsilon _{1}zW_{0})}{\Phi},$

Symbols:: $z$ : real, $\varepsilon _{j}=\pm 1$ , $\Phi$ , $\mathcal{T}_{j}$ : transformation and $W_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E29
Encodings:: TeX, pMML, png

provided that the numerator on the right-hand side does not vanish. Again, since $\varepsilon _{j}=\pm 1$ , $j=1,2,3$ , independently, there are eight distinct transformations of type $\mathcal{T}_{{\varepsilon _{1},\varepsilon _{2},\varepsilon _{3}}}$ .

§32.7(vi) Relationship Between the Third and Fifth Painlevé Equations

Notes:: See Gromak (1975); also Gromak et al. (2002, §34).
Permalink:: http://dlmf.nist.gov/32.7.vi

Let $w=w(z;\alpha,\beta,1,-1)$ be a solution of $\mbox{P}_{{\mbox{\scriptsize III}}}$ and

32.7.30 $v=w^{{\prime}}-\varepsilon w^{2}+(\ifrac{(1-\varepsilon\alpha)w}{z}),$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $v$ and $\varepsilon=\pm 1$
Permalink:: http://dlmf.nist.gov/32.7.E30
Encodings:: TeX, pMML, png

with $\varepsilon=\pm 1$ . Then

32.7.31

$W(\zeta;\alpha _{0},\beta _{0},\gamma _{0},\delta _{0})=\frac{v-1}{v+1},$

$z=\sqrt{2\zeta},$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant, $\delta$ : arbitrary constant, $v$ and $W(\zeta,\varepsilon/2)$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E31
Encodings:: TeX, TeX, pMML, pMML, png, png

satisfies $\mbox{P}_{{\mbox{\scriptsize V}}}$ with

32.7.32 $(\alpha _{0},\beta _{0},\gamma _{0},\delta _{0})={\left((\beta-\varepsilon\alpha+2)^{2}/32,-(\beta+\varepsilon\alpha-2)^{2}/32,-\varepsilon,0\right)}.$

Symbols:: $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant, $\delta$ : arbitrary constant and $\varepsilon=\pm 1$
Permalink:: http://dlmf.nist.gov/32.7.E32
Encodings:: TeX, pMML, png

§32.7(vii) Sixth Painlevé Equation

Notes:: See Okamoto (1987a); also Cosgrove (2006), Gromak et al. (2002, §§42,47).
Keywords:: Painlevé equations
Permalink:: http://dlmf.nist.gov/32.7.vii

Let $w_{j}(z_{j})=w_{j}(z_{j};\alpha _{j},\beta _{j},\gamma _{j},\delta _{j})$ , $j=0,1,2,3$ , be solutions of $\mbox{P}_{{\mbox{\scriptsize VI}}}$ with

32.7.33 $z_{1}=1/z_{0},$

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.7.E33
Encodings:: TeX, pMML, png

32.7.34 $z_{2}=1-z_{0},$

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.7.E34
Encodings:: TeX, pMML, png

32.7.35 $z_{3}=1/z_{0},$

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.7.E35
Encodings:: TeX, pMML, png

32.7.36 $(\alpha _{1},\beta _{1},\gamma _{1},\delta _{1})=(\alpha _{0},\beta _{0},-\delta _{0}+\tfrac{1}{2},-\gamma _{0}+\tfrac{1}{2}),$

Symbols:: $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.7.E36
Encodings:: TeX, pMML, png

32.7.37 $(\alpha _{2},\beta _{2},\gamma _{2},\delta _{2})=(\alpha _{0},-\gamma _{0},-\beta _{0},\delta _{0}),$

Symbols:: $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.7.E37
Encodings:: TeX, pMML, png

32.7.38 $(\alpha _{3},\beta _{3},\gamma _{3},\delta _{3})=(-\beta _{0},-\alpha _{0},\gamma _{0},\delta _{0}).$

Symbols:: $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.7.E38
Encodings:: TeX, pMML, png

Then

32.7.39 $\mathcal{S}_{1}:\enskip w_{1}(z_{1})=w_{0}(z_{0})/z_{0},$

Symbols:: $z$ : real and $\mathcal{S}_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E39
Encodings:: TeX, pMML, png

32.7.40 $\mathcal{S}_{2}:\enskip w_{2}(z_{2})=1-w_{0}(z_{0}),$

Symbols:: $z$ : real and $\mathcal{S}_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E40
Encodings:: TeX, pMML, png

32.7.41 $\mathcal{S}_{3}:\enskip w_{3}(z_{3})=1/w_{0}(z_{0}).$

Symbols:: $z$ : real and $\mathcal{S}_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E41
Encodings:: TeX, pMML, png

The transformations $\mathcal{S}_{j}$ , for $j=1,2,3$ , generate a group of order 24. See Iwasaki et al. (1991, p. 127).

Let $w(z;\alpha,\beta,\gamma,\delta)$ and $W(z;A,B,C,D)$ be solutions of $\mbox{P}_{{\mbox{\scriptsize VI}}}$ with

32.7.42 $(\alpha,\beta,\gamma,\delta)=\left(\tfrac{1}{2}(\theta _{\infty}-1)^{2},-\tfrac{1}{2}\theta _{0}^{2},\tfrac{1}{2}\theta _{1}^{2},\tfrac{1}{2}(1-\theta _{2}^{2})\right),$

Symbols:: $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant, $\delta$ : arbitrary constant and $\theta _{j}$ : parameters
Permalink:: http://dlmf.nist.gov/32.7.E42
Encodings:: TeX, pMML, png

32.7.43 $(A,B,C,D)=\left(\tfrac{1}{2}(\Theta _{\infty}-1)^{2},-\tfrac{1}{2}\Theta _{0}^{2},\tfrac{1}{2}\Theta _{1}^{2},\tfrac{1}{2}(1-\Theta _{2}^{2})\right),$

Symbols:: $\Theta _{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E43
Encodings:: TeX, pMML, png

and

32.7.44 $\theta _{j}=\Theta _{j}+\tfrac{1}{2}\sigma,$

Symbols:: $\Theta _{j}$ : transformation, $\theta _{j}$ : parameters and $\sigma$
Permalink:: http://dlmf.nist.gov/32.7.E44
Encodings:: TeX, pMML, png

for $j=0,1,2,\infty$ , where

32.7.45 $\sigma=\theta _{0}+\theta _{1}+\theta _{2}+\theta _{{\infty}}-1=1-(\Theta _{0}+\Theta _{1}+\Theta _{2}+\Theta _{{\infty}}).$

Symbols:: $\Theta _{j}$ : transformation, $\theta _{j}$ : parameters and $\sigma$
Permalink:: http://dlmf.nist.gov/32.7.E45
Encodings:: TeX, pMML, png

Then

32.7.46 $\frac{\sigma}{w-W}=\frac{z(z-1)W^{{\prime}}}{W(W-1)(W-z)}+\frac{\Theta _{0}}{W}+\frac{\Theta _{1}}{W-1}+\frac{\Theta _{2}-1}{W-z}=\frac{z(z-1)w^{{\prime}}}{w(w-1)(w-z)}+\frac{\theta _{0}}{w}+\frac{\theta _{1}}{w-1}+\frac{\theta _{2}-1}{w-z}.$

Symbols:: $z$ : real, $\Theta _{j}$ : transformation, $\theta _{j}$ : parameters, $\sigma$ and $W(\zeta,\varepsilon/2)$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E46
Encodings:: TeX, pMML, png

$\mbox{P}_{{\mbox{\scriptsize VI}}}$ also has quadratic and quartic transformations. Let $w=w(z;\alpha,\beta,\gamma,\delta)$ be a solution of $\mbox{P}_{{\mbox{\scriptsize VI}}}$ . The quadratic transformation

32.7.47

$u_{1}(\zeta _{1})=\frac{(1-w)(w-z)}{(1+\sqrt{z})^{2}w},$

$\zeta _{1}=\left(\frac{1-\sqrt{z}}{1+\sqrt{z}}\right)^{2},$

Symbols:: $z$ : real, $u_{1}$ : quadratic transformation and $\zeta _{1}$ : quadratic transformation
Permalink:: http://dlmf.nist.gov/32.7.E47
Encodings:: TeX, TeX, pMML, pMML, png, png

transforms $\mbox{P}_{{\mbox{\scriptsize VI}}}$ with $\alpha=-\beta$ and $\gamma=\tfrac{1}{2}-\delta$ to $\mbox{P}_{{\mbox{\scriptsize VI}}}$ with $(\alpha _{1},\beta _{1},\gamma _{1},\delta _{1})=(4\alpha,-4\gamma,0,\tfrac{1}{2})$ . The quartic transformation

32.7.48

$u_{2}(\zeta _{2})=\frac{(w^{2}-z)^{2}}{4w(w-1)(w-z)},$

$\zeta _{2}=z,$

Symbols:: $z$ : real, $u_{2}$ : quartic transformation and $\zeta _{2}$ : quartic transformation
Permalink:: http://dlmf.nist.gov/32.7.E48
Encodings:: TeX, TeX, pMML, pMML, png, png

transforms $\mbox{P}_{{\mbox{\scriptsize VI}}}$ with $\alpha=-\beta=\gamma=\tfrac{1}{2}-\delta$ to $\mbox{P}_{{\mbox{\scriptsize VI}}}$ with $(\alpha _{2},\beta _{2},\gamma _{2},\delta _{2})=(16\alpha,0,0,\tfrac{1}{2})$ . Also,

32.7.49 $u_{3}(\zeta _{3})=\left(\frac{1-z^{{1/4}}}{1+z^{{1/4}}}\right)^{2}\left(\frac{\sqrt{w}+z^{{1/4}}}{\sqrt{w}-z^{{1/4}}}\right)^{2},$

Symbols:: $z$ : real, $u_{3}$ : transformation and $\zeta _{3}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E49
Encodings:: TeX, pMML, png

32.7.50 $\zeta _{3}=\left(\frac{1-z^{{1/4}}}{1+z^{{1/4}}}\right)^{4},$

Symbols:: $z$ : real and $\zeta _{3}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E50
Encodings:: TeX, pMML, png

transforms $\mbox{P}_{{\mbox{\scriptsize VI}}}$ with $\alpha=\beta=0$ and $\gamma=\tfrac{1}{2}-\delta$ to $\mbox{P}_{{\mbox{\scriptsize VI}}}$ with $\alpha _{3}=\beta _{3}$ and $\gamma _{3}=\tfrac{1}{2}-\delta _{3}$ .