# §32.7 Bäcklund Transformations

## §32.7(i) Definition

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

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 See also: Annotations for §32.7(ii), §32.7 and Ch.32

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 See also: Annotations for §32.7(ii), §32.7 and Ch.32

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

or equivalently,

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

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 also: Annotations for §32.7(ii), §32.7 and Ch.32

See Fokas et al. (1993).

## §32.7(iii) Third Painlevé Equation

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 See also: Annotations for §32.7(iii), §32.7 and Ch.32
 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 See also: Annotations for §32.7(iii), §32.7 and Ch.32

Then

 32.7.8 $\displaystyle\mathcal{S}_{1}:\enskip w_{1}$ $\displaystyle=-w_{0},$ ⓘ Symbols: $\mathcal{S}_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E8 Encodings: TeX, pMML, png See also: Annotations for §32.7(iii), §32.7 and Ch.32 32.7.9 $\displaystyle\mathcal{S}_{2}:\enskip w_{2}$ $\displaystyle=\ifrac{1}{w_{0}}.$ ⓘ Symbols: $\mathcal{S}_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E9 Encodings: TeX, pMML, png See also: Annotations for §32.7(iii), §32.7 and Ch.32

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 $\displaystyle\alpha_{1}$ $\displaystyle=\alpha_{3}=\alpha_{0}+2,$ $\displaystyle\alpha_{2}$ $\displaystyle=\alpha_{4}=\alpha_{0}-2,$ $\displaystyle\beta_{1}$ $\displaystyle=\beta_{2}=\beta_{0}+2,$ $\displaystyle\beta_{3}$ $\displaystyle=\beta_{4}=\beta_{0}-2.$ ⓘ 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 See also: Annotations for §32.7(iii), §32.7 and Ch.32

Then

 32.7.11 $\displaystyle\mathcal{T}_{1}:\enskip W_{1}$ $\displaystyle=\frac{zW_{0}^{\prime}+zW_{0}^{2}-\beta W_{0}-W_{0}+z}{W_{0}(zW_{% 0}^{\prime}+zW_{0}^{2}+\alpha W_{0}+W_{0}+z)},$ 32.7.12 $\displaystyle\mathcal{T}_{2}:\enskip W_{2}$ $\displaystyle=-\frac{zW_{0}^{\prime}-zW_{0}^{2}-\beta W_{0}-W_{0}+z}{W_{0}(zW_% {0}^{\prime}-zW_{0}^{2}-\alpha W_{0}+W_{0}+z)},$ 32.7.13 $\displaystyle\mathcal{T}_{3}:\enskip W_{3}$ $\displaystyle=-\frac{zW_{0}^{\prime}+zW_{0}^{2}+\beta W_{0}-W_{0}-z}{W_{0}(zW_% {0}^{\prime}+zW_{0}^{2}+\alpha W_{0}+W_{0}-z)},$ 32.7.14 $\displaystyle\mathcal{T}_{4}:\enskip W_{4}$ $\displaystyle=\frac{zW_{0}^{\prime}-zW_{0}^{2}+\beta W_{0}-W_{0}-z}{W_{0}(zW_{% 0}^{\prime}-zW_{0}^{2}-\alpha W_{0}+W_{0}-z)}.$

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 $\displaystyle\beta_{5}$ $\displaystyle=\beta_{0}+2,$ $\displaystyle\beta_{6}$ $\displaystyle=\beta_{0}-2.$ ⓘ Symbols: $\beta$: arbitrary constant Permalink: http://dlmf.nist.gov/32.7.E15 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §32.7(iii), §32.7 and Ch.32

Then

 32.7.16 $\displaystyle\mathcal{T}_{5}:\enskip u_{5}$ $\displaystyle=\ifrac{(zu_{0}^{\prime}+z-(\beta_{0}+1)u_{0})}{u_{0}^{2}},$ ⓘ 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 See also: Annotations for §32.7(iii), §32.7 and Ch.32 32.7.17 $\displaystyle\mathcal{T}_{6}:\enskip u_{6}$ $\displaystyle=-\ifrac{(zu_{0}^{\prime}-z+(\beta_{0}-1)u_{0})}{u_{0}^{2}}.$ ⓘ 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 See also: Annotations for §32.7(iii), §32.7 and Ch.32

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

Furthermore,

 32.7.18 $\displaystyle w(z;a,b,0,0)$ $\displaystyle=W^{2}(\zeta;0,0,a,b),$ $\displaystyle z$ $\displaystyle=\tfrac{1}{2}\zeta^{2}.$ ⓘ Symbols: $z$: real and $W(\zeta,\varepsilon/2)$: transformation Permalink: http://dlmf.nist.gov/32.7.E18 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §32.7(iii), §32.7 and Ch.32

## §32.7(iv) Fourth Painlevé Equation

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 $\displaystyle\alpha_{1}^{\pm}$ $\displaystyle=\tfrac{1}{4}\left(2-2\alpha_{0}\pm 3\sqrt{-2\beta_{0}}\right),$ $\displaystyle\beta_{1}^{\pm}$ $\displaystyle=-\tfrac{1}{2}\left(1+\alpha_{0}\pm\tfrac{1}{2}\sqrt{-2\beta_{0}}% \right)^{2},$ $\displaystyle\alpha_{2}^{\pm}$ $\displaystyle=-\tfrac{1}{4}\left(2+2\alpha_{0}\pm 3\sqrt{-2\beta_{0}}\right),$ $\displaystyle\beta_{2}^{\pm}$ $\displaystyle=-\tfrac{1}{2}\left(1-\alpha_{0}\pm\tfrac{1}{2}\sqrt{-2\beta_{0}}% \right)^{2},$ $\displaystyle\alpha_{3}^{\pm}$ $\displaystyle=\tfrac{3}{2}-\tfrac{1}{2}\alpha_{0}\mp\tfrac{3}{4}\sqrt{-2\beta_% {0}},$ $\displaystyle\beta_{3}^{\pm}$ $\displaystyle=-\tfrac{1}{2}\left(1-\alpha_{0}\pm\tfrac{1}{2}\sqrt{-2\beta_{0}}% \right)^{2},$ $\displaystyle\alpha_{4}^{\pm}$ $\displaystyle=-\tfrac{3}{2}-\tfrac{1}{2}\alpha_{0}\mp\tfrac{3}{4}\sqrt{-2\beta% _{0}},$ $\displaystyle\beta_{4}^{\pm}$ $\displaystyle=-\tfrac{1}{2}\left(-1-\alpha_{0}\pm\tfrac{1}{2}\sqrt{-2\beta_{0}% }\right)^{2}.$ ⓘ 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 See also: Annotations for §32.7(iv), §32.7 and Ch.32

Then

 32.7.20 $\displaystyle\mathcal{T}_{1}^{\pm}:\enskip w_{1}^{\pm}$ $\displaystyle=\frac{w_{0}^{\prime}-w_{0}^{2}-2zw_{0}\mp\sqrt{-2\beta_{0}}}{2w_% {0}},$ ⓘ Symbols: $z$: real, $\beta$: arbitrary constant and $\mathcal{T}_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E20 Encodings: TeX, pMML, png See also: Annotations for §32.7(iv), §32.7 and Ch.32 32.7.21 $\displaystyle\mathcal{T}_{2}^{\pm}:\enskip w_{2}^{\pm}$ $\displaystyle=-\frac{w_{0}^{\prime}+w_{0}^{2}+2zw_{0}\mp\sqrt{-2\beta_{0}}}{2w% _{0}},$ ⓘ Symbols: $z$: real, $\beta$: arbitrary constant and $\mathcal{T}_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E21 Encodings: TeX, pMML, png See also: Annotations for §32.7(iv), §32.7 and Ch.32 32.7.22 $\displaystyle\mathcal{T}_{3}^{\pm}:\enskip w_{3}^{\pm}$ $\displaystyle=w_{0}+\frac{2\left(1-\alpha_{0}\mp\tfrac{1}{2}\sqrt{-2\beta_{0}}% \right)w_{0}}{w_{0}^{\prime}\pm\sqrt{-2\beta_{0}}+2zw_{0}+w_{0}^{2}},$ ⓘ 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 See also: Annotations for §32.7(iv), §32.7 and Ch.32 32.7.23 $\displaystyle\mathcal{T}_{4}^{\pm}:\enskip w_{4}^{\pm}$ $\displaystyle=w_{0}+\frac{2\left(1+\alpha_{0}\pm\tfrac{1}{2}\sqrt{-2\beta_{0}}% \right)w_{0}}{w_{0}^{\prime}\mp\sqrt{-2\beta_{0}}-2zw_{0}-w_{0}^{2}},$ ⓘ 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 See also: Annotations for §32.7(iv), §32.7 and Ch.32

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

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 $\displaystyle z_{1}$ $\displaystyle=-z_{0},$ $\displaystyle z_{2}$ $\displaystyle=z_{0},$ $\displaystyle(\alpha_{1},\beta_{1},\gamma_{1},\delta_{1})$ $\displaystyle=(\alpha_{0},\beta_{0},-\gamma_{0},\delta_{0}),$ $\displaystyle(\alpha_{2},\beta_{2},\gamma_{2},\delta_{2})$ $\displaystyle=(-\beta_{0},-\alpha_{0},-\gamma_{0},\delta_{0}).$ ⓘ 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 See also: Annotations for §32.7(v), §32.7 and Ch.32

Then

 32.7.25 $\displaystyle\mathcal{S}_{1}:\enskip w_{1}(z_{1})$ $\displaystyle=w(z_{0}),$ ⓘ Symbols: $z$: real and $\mathcal{S}_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E25 Encodings: TeX, pMML, png See also: Annotations for §32.7(v), §32.7 and Ch.32 32.7.26 $\displaystyle\mathcal{S}_{2}:\enskip w_{2}(z_{2})$ $\displaystyle=\ifrac{1}{w(z_{0})}.$ ⓘ Symbols: $z$: real and $\mathcal{S}_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E26 Encodings: TeX, pMML, png See also: Annotations for §32.7(v), §32.7 and Ch.32

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 $\displaystyle\alpha_{1}$ $\displaystyle=\tfrac{1}{8}\left(\gamma_{0}+\varepsilon_{1}\left(1-\varepsilon_% {3}\sqrt{-2\beta_{0}}-\varepsilon_{2}\sqrt{2\alpha_{0}}\right)\right)^{2},$ $\displaystyle\beta_{1}$ $\displaystyle=-\tfrac{1}{8}\left(\gamma_{0}-\varepsilon_{1}\left(1-\varepsilon% _{3}\sqrt{-2\beta_{0}}-\varepsilon_{2}\sqrt{2\alpha_{0}}\right)\right)^{2},$ $\displaystyle\gamma_{1}$ $\displaystyle=\varepsilon_{1}\left(\varepsilon_{3}\sqrt{-2\beta_{0}}-% \varepsilon_{2}\sqrt{2\alpha_{0}}\right),$ ⓘ 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 See also: Annotations for §32.7(v), §32.7 and Ch.32

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

 32.7.28 $\Phi=zW_{0}^{\prime}-\varepsilon_{2}\sqrt{2\alpha_{0}}W_{0}^{2}+\varepsilon_{3% }\sqrt{-2\beta_{0}}+\left(\varepsilon_{2}\sqrt{2\alpha_{0}}-\varepsilon_{3}% \sqrt{-2\beta_{0}}+\varepsilon_{1}z\right)W_{0},$

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 See also: Annotations for §32.7(v), §32.7 and Ch.32

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

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 See also: Annotations for §32.7(vi), §32.7 and Ch.32

with $\varepsilon=\pm 1$. Then

 32.7.31 $\displaystyle W(\zeta;\alpha_{0},\beta_{0},\gamma_{0},\delta_{0})$ $\displaystyle=\frac{v-1}{v+1},$ $\displaystyle z$ $\displaystyle=\sqrt{2\zeta},$

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

 32.7.32 $(\alpha_{0},\beta_{0},\gamma_{0},\delta_{0})={\left((\beta-\varepsilon\alpha+2% )^{2}/32,-(\beta+\varepsilon\alpha-2)^{2}/32,-\varepsilon,0\right)}.$

## §32.7(vii) Sixth Painlevé Equation

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 $\displaystyle z_{1}$ $\displaystyle=1/z_{0},$ ⓘ Symbols: $z$: real Permalink: http://dlmf.nist.gov/32.7.E33 Encodings: TeX, pMML, png See also: Annotations for §32.7(vii), §32.7 and Ch.32 32.7.34 $\displaystyle z_{2}$ $\displaystyle=1-z_{0},$ ⓘ Symbols: $z$: real Permalink: http://dlmf.nist.gov/32.7.E34 Encodings: TeX, pMML, png See also: Annotations for §32.7(vii), §32.7 and Ch.32 32.7.35 $\displaystyle z_{3}$ $\displaystyle=1/z_{0},$ ⓘ Symbols: $z$: real Permalink: http://dlmf.nist.gov/32.7.E35 Encodings: TeX, pMML, png See also: Annotations for §32.7(vii), §32.7 and Ch.32
 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 See also: Annotations for §32.7(vii), §32.7 and Ch.32
 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 See also: Annotations for §32.7(vii), §32.7 and Ch.32
 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 See also: Annotations for §32.7(vii), §32.7 and Ch.32

Then

 32.7.39 $\displaystyle\mathcal{S}_{1}:\enskip w_{1}(z_{1})$ $\displaystyle=w_{0}(z_{0})/z_{0},$ ⓘ Symbols: $z$: real and $\mathcal{S}_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E39 Encodings: TeX, pMML, png See also: Annotations for §32.7(vii), §32.7 and Ch.32 32.7.40 $\displaystyle\mathcal{S}_{2}:\enskip w_{2}(z_{2})$ $\displaystyle=1-w_{0}(z_{0}),$ ⓘ Symbols: $z$: real and $\mathcal{S}_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E40 Encodings: TeX, pMML, png See also: Annotations for §32.7(vii), §32.7 and Ch.32 32.7.41 $\displaystyle\mathcal{S}_{3}:\enskip w_{3}(z_{3})$ $\displaystyle=1/w_{0}(z_{0}).$ ⓘ Symbols: $z$: real and $\mathcal{S}_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E41 Encodings: TeX, pMML, png See also: Annotations for §32.7(vii), §32.7 and Ch.32

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 $(\alpha,\beta,\gamma,\delta)=\left(\tfrac{1}{2}(\theta_{\infty}-1)^{2},-\tfrac% {1}{2}\theta_{0}^{2},\tfrac{1}{2}\theta_{1}^{2},\tfrac{1}{2}(1-\theta_{2}^{2})% \right),$
 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 See also: Annotations for §32.7(vii), §32.7 and Ch.32

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 See also: Annotations for §32.7(vii), §32.7 and Ch.32

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 See also: Annotations for §32.7(vii), §32.7 and Ch.32

Then

 32.7.46 $\frac{\sigma}{w-W}=\frac{z(z-1)W^{\prime}}{W(W-1)(W-z)}+\frac{\Theta_{0}}{W}+% \frac{\Theta_{1}}{W-1}+\frac{\Theta_{2}-1}{W-z}=\frac{z(z-1)w^{\prime}}{w(w-1)% (w-z)}+\frac{\theta_{0}}{w}+\frac{\theta_{1}}{w-1}+\frac{\theta_{2}-1}{w-z}.$ ⓘ

$\mbox{P}_{\mbox{\scriptsize VI}}$ 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 $\displaystyle u_{1}(\zeta_{1})$ $\displaystyle=\frac{(1-w)(w-z)}{(1+\sqrt{z})^{2}w},$ $\displaystyle\zeta_{1}$ $\displaystyle=\left(\frac{1-\sqrt{z}}{1+\sqrt{z}}\right)^{2},$ ⓘ 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 See also: Annotations for §32.7(vii), §32.7 and Ch.32

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 $\displaystyle u_{2}(\zeta_{2})$ $\displaystyle=\frac{(w^{2}-z)^{2}}{4w(w-1)(w-z)},$ $\displaystyle\zeta_{2}$ $\displaystyle=z,$ ⓘ 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 See also: Annotations for §32.7(vii), §32.7 and Ch.32

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 See also: Annotations for §32.7(vii), §32.7 and Ch.32
 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 See also: Annotations for §32.7(vii), §32.7 and Ch.32

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 $\gamma_{3}=\tfrac{1}{2}-\delta_{3}$.

## §32.7(viii) Affine Weyl Groups

See Okamoto (1986, 1987a, 1987b, 1987c), Sakai (2001), Umemura (2000).