§32.7 Bäcklund Transformations

With the exception of ${\text{P}}_{\text{I}}$, 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.

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

32.7.1		$$𝒮:w(z;-\alpha )=-w,$$
ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant and $𝒮$: 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		$${𝒯}^{\pm }:w(z;\alpha \pm 1)=-w-\frac{2\alpha \pm 1}{2w^2\pm 2w^{\prime }+z},$$
ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant and ${𝒯}^{\pm }$: 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 ${\text{P}}_{\text{II}}$, provided that $\alpha \ne \mp \frac{1}{2}$. ${\text{P}}_{\text{II}}$ also has the special transformation

32.7.3		$$W(\zeta ;\frac{1}{2}\epsilon )=\frac{2^{-1/3}\epsilon }{w(z;0)}\frac{d}{dz}w(z;0),$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $z$: real, $W(\zeta ,\epsilon /2)$: transformation and $\epsilon =\pm 1$ Permalink: http://dlmf.nist.gov/32.7.E3 Encodings: TeX, pMML, png See also: Annotations for §32.7(ii), §32.7 and Ch.32

or equivalently,

32.7.4		$$w^2(z;0)=2^{-1/3}\left(W^2(\zeta ;\frac{1}{2}\epsilon )-\epsilon \frac{d}{d\zeta }W(\zeta ;\frac{1}{2}\epsilon )+\frac{1}{2}\zeta \right),$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $z$: real, $W(\zeta ,\epsilon /2)$: transformation and $\epsilon =\pm 1$ Permalink: http://dlmf.nist.gov/32.7.E4 Encodings: TeX, pMML, png See also: Annotations for §32.7(ii), §32.7 and Ch.32

with $\zeta =-2^{1/3}z$ and $\epsilon =\pm 1$, where $W(\zeta ;\frac{1}{2}\epsilon )$ satisfies ${\text{P}}_{\text{II}}$ with $z=\zeta$, $\alpha =\frac{1}{2}\epsilon$, and $w(z;0)$ satisfies ${\text{P}}_{\text{II}}$ 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		$$\frac{\alpha +\frac{1}{2}}{w_{\alpha +1}+w_{\alpha }}+\frac{\alpha -\frac{1}{2}}{w_{\alpha }+w_{\alpha -1}}+2w_{\alpha }^2+z=0.$$
ⓘ 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).

Let $w_j=w(z;{\alpha }_j,{\beta }_j,{\gamma }_j,{\delta }_j)$, $j=0,1,2$, be solutions of ${\text{P}}_{\text{III}}$ 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		${𝒮}_1:w_1$	$=-w_0,$
ⓘ Symbols: ${𝒮}_j$: 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		${𝒮}_2:w_2$	$=1/w_0.$
ⓘ Symbols: ${𝒮}_j$: 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 ${\text{P}}_{\text{III}}$ 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 See also: Annotations for §32.7(iii), §32.7 and Ch.32

Then

32.7.11		${𝒯}_1:W_1$	$={\displaystyle \frac{z{W}_0^{\prime }+z{W}_0^2-\beta W_0-W_0+z}{W_0(z{W}_0^{\prime }+z{W}_0^2+\alpha W_0+W_0+z)}},$
ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $W_j$: transformation and ${𝒯}_j$: transformation Permalink: http://dlmf.nist.gov/32.7.E11 Encodings: TeX, pMML, png See also: Annotations for §32.7(iii), §32.7 and Ch.32
32.7.12		${𝒯}_2:W_2$	$=-{\displaystyle \frac{z{W}_0^{\prime }-z{W}_0^2-\beta W_0-W_0+z}{W_0(z{W}_0^{\prime }-z{W}_0^2-\alpha W_0+W_0+z)}},$
ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $W_j$: transformation and ${𝒯}_j$: transformation Permalink: http://dlmf.nist.gov/32.7.E12 Encodings: TeX, pMML, png See also: Annotations for §32.7(iii), §32.7 and Ch.32
32.7.13		${𝒯}_3:W_3$	$=-{\displaystyle \frac{z{W}_0^{\prime }+z{W}_0^2+\beta W_0-W_0-z}{W_0(z{W}_0^{\prime }+z{W}_0^2+\alpha W_0+W_0-z)}},$
ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $W_j$: transformation and ${𝒯}_j$: transformation Permalink: http://dlmf.nist.gov/32.7.E13 Encodings: TeX, pMML, png See also: Annotations for §32.7(iii), §32.7 and Ch.32
32.7.14		${𝒯}_4:W_4$	$={\displaystyle \frac{z{W}_0^{\prime }-z{W}_0^2+\beta W_0-W_0-z}{W_0(z{W}_0^{\prime }-z{W}_0^2-\alpha W_0+W_0-z)}}.$
ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $W_j$: transformation and ${𝒯}_j$: transformation Permalink: http://dlmf.nist.gov/32.7.E14 Encodings: TeX, pMML, png See also: Annotations for §32.7(iii), §32.7 and Ch.32

See Milne et al. (1997).

If $\gamma =0$ and $\alpha \delta \ne 0$, 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 ${\text{P}}_{\text{III}}$ 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 See also: Annotations for §32.7(iii), §32.7 and Ch.32

Then

32.7.16		${𝒯}_5:u_5$	$=(z{u}_0^{\prime }+z-({\beta }_0+1)u_0)/u_0^2,$
ⓘ Symbols: $z$: real, $\beta$: arbitrary constant, ${𝒯}_j$: 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		${𝒯}_6:u_6$	$=-(z{u}_0^{\prime }-z+({\beta }_0-1)u_0)/u_0^2.$
ⓘ Symbols: $z$: real, $\beta$: arbitrary constant, ${𝒯}_j$: 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 ${\text{P}}_{\text{III}}$ with $\delta =0$ and $\beta \gamma \ne 0$.

Furthermore,

32.7.18		$w(z;a,b,0,0)$	$=W^2(\zeta ;0,0,a,b),$
		$z$	$=\frac{1}{2}{\zeta }^2.$
ⓘ Symbols: $z$: real and $W(\zeta ,\epsilon /2)$: 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

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 ${\text{P}}_{\text{IV}}$ with

32.7.19		${\alpha }_1^{\pm }$	$=\frac{1}{4}\left(2-2{\alpha }_0\pm 3\sqrt{-2{\beta }_0}\right),$
		${\beta }_1^{\pm }$	$=-\frac{1}{2}{\left(1+{\alpha }_0\pm \frac{1}{2}\sqrt{-2{\beta }_0}\right)}^2,$
		${\alpha }_2^{\pm }$	$=-\frac{1}{4}\left(2+2{\alpha }_0\pm 3\sqrt{-2{\beta }_0}\right),$
		${\beta }_2^{\pm }$	$=-\frac{1}{2}{\left(1-{\alpha }_0\pm \frac{1}{2}\sqrt{-2{\beta }_0}\right)}^2,$
		${\alpha }_3^{\pm }$	$=\frac{3}{2}-\frac{1}{2}{\alpha }_0\mp \frac{3}{4}\sqrt{-2{\beta }_0},$
		${\beta }_3^{\pm }$	$=-\frac{1}{2}{\left(1-{\alpha }_0\pm \frac{1}{2}\sqrt{-2{\beta }_0}\right)}^2,$
		${\alpha }_4^{\pm }$	$=-\frac{3}{2}-\frac{1}{2}{\alpha }_0\mp \frac{3}{4}\sqrt{-2{\beta }_0},$
		${\beta }_4^{\pm }$	$=-\frac{1}{2}{\left(-1-{\alpha }_0\pm \frac{1}{2}\sqrt{-2{\beta }_0}\right)}^2.$
ⓘ 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		${𝒯}_1^{\pm }:w_1^{\pm }$	$={\displaystyle \frac{w_0^{\prime }-w_0^2-2z{w}_0\mp \sqrt{-2{\beta }_0}}{2w_0}},$
ⓘ Symbols: $z$: real, $\beta$: arbitrary constant and ${𝒯}_j$: 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		${𝒯}_2^{\pm }:w_2^{\pm }$	$=-{\displaystyle \frac{w_0^{\prime }+w_0^2+2z{w}_0\mp \sqrt{-2{\beta }_0}}{2w_0}},$
ⓘ Symbols: $z$: real, $\beta$: arbitrary constant and ${𝒯}_j$: 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		${𝒯}_3^{\pm }:w_3^{\pm }$	$=w_0+{\displaystyle \frac{2\left(1-{\alpha }_0\mp \frac{1}{2}\sqrt{-2{\beta }_0}\right)w_0}{w_0^{\prime }\pm \sqrt{-2{\beta }_0}+2z{w}_0+w_0^2}},$
ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant and ${𝒯}_j$: 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		${𝒯}_4^{\pm }:w_4^{\pm }$	$=w_0+{\displaystyle \frac{2\left(1+{\alpha }_0\pm \frac{1}{2}\sqrt{-2{\beta }_0}\right)w_0}{w_0^{\prime }\mp \sqrt{-2{\beta }_0}-2z{w}_0-w_0^2}},$
ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant and ${𝒯}_j$: 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).

Let $w_j(z_j)=w(z_j;{\alpha }_j,{\beta }_j,{\gamma }_j,{\delta }_j)$, $j=0,1,2$, be solutions of ${\text{P}}_{\text{V}}$ 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 See also: Annotations for §32.7(v), §32.7 and Ch.32

Then

32.7.25		${𝒮}_1:w_1(z_1)$	$=w(z_0),$
ⓘ Symbols: $z$: real and ${𝒮}_j$: 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		${𝒮}_2:w_2(z_2)$	$=1/w(z_0).$
ⓘ Symbols: $z$: real and ${𝒮}_j$: 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 $W_0=W(z;{\alpha }_0,{\beta }_0,{\gamma }_0,-\frac{1}{2})$ and $W_1=W(z;{\alpha }_1,{\beta }_1,{\gamma }_1,-\frac{1}{2})$ be solutions of ${\text{P}}_{\text{V}}$, where

32.7.27		${\alpha }_1$	$=\frac{1}{8}{\left({\gamma }_0+{\epsilon }_1\left(1-{\epsilon }_3\sqrt{-2{\beta }_0}-{\epsilon }_2\sqrt{2{\alpha }_0}\right)\right)}^2,$
		${\beta }_1$	$=-\frac{1}{8}{\left({\gamma }_0-{\epsilon }_1\left(1-{\epsilon }_3\sqrt{-2{\beta }_0}-{\epsilon }_2\sqrt{2{\alpha }_0}\right)\right)}^2,$
		${\gamma }_1$	$={\epsilon }_1\left({\epsilon }_3\sqrt{-2{\beta }_0}-{\epsilon }_2\sqrt{2{\alpha }_0}\right),$
ⓘ Symbols: $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant and ${\epsilon }_j=\pm 1$ 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 ${\epsilon }_j=\pm 1$, $j=1,2,3$, independently. Also let

32.7.28		$$\mathrm{\Phi }=z{W}_0^{\prime }-{\epsilon }_2\sqrt{2{\alpha }_0}{W}_0^2+{\epsilon }_3\sqrt{-2{\beta }_0}+\left({\epsilon }_2\sqrt{2{\alpha }_0}-{\epsilon }_3\sqrt{-2{\beta }_0}+{\epsilon }_{1}z\right)W_0,$$
ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, ${\epsilon }_j=\pm 1$, $\mathrm{\Phi }$ and $W_j$: transformation Permalink: http://dlmf.nist.gov/32.7.E28 Encodings: TeX, pMML, png See also: Annotations for §32.7(v), §32.7 and Ch.32

and assume $\mathrm{\Phi }\ne 0$. Then

32.7.29		$${𝒯}_{{\epsilon }_1,{\epsilon }_2,{\epsilon }_3}:W_1=(\mathrm{\Phi }-2{\epsilon }_{1}z{W}_0)/\mathrm{\Phi },$$
ⓘ Symbols: $z$: real, ${\epsilon }_j=\pm 1$, $\mathrm{\Phi }$, ${𝒯}_j$: 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 ${\epsilon }_j=\pm 1$, $j=1,2,3$, independently, there are eight distinct transformations of type ${𝒯}_{{\epsilon }_1,{\epsilon }_2,{\epsilon }_3}$.

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

32.7.30		$$v=w^{\prime }-\epsilon w^2+((1-\epsilon \alpha )w/z),$$
ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $v$ and $\epsilon =\pm 1$ 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 $\epsilon =\pm 1$. Then

32.7.31		$W(\zeta ;{\alpha }_0,{\beta }_0,{\gamma }_0,{\delta }_0)$	$={\displaystyle \frac{v-1}{v+1}},$
		$z$	$=\sqrt{2\zeta },$
ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant, $\delta$: arbitrary constant, $v$ and $W(\zeta ,\epsilon /2)$: transformation Permalink: http://dlmf.nist.gov/32.7.E31 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §32.7(vi), §32.7 and Ch.32

satisfies ${\text{P}}_{\text{V}}$ with

32.7.32		$$({\alpha }_0,{\beta }_0,{\gamma }_0,{\delta }_0)=({(\beta -\epsilon \alpha +2)}^2/32,-{(\beta +\epsilon \alpha -2)}^2/32,-\epsilon ,0).$$
ⓘ Symbols: $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant, $\delta$: arbitrary constant and $\epsilon =\pm 1$ Permalink: http://dlmf.nist.gov/32.7.E32 Encodings: TeX, pMML, png See also: Annotations for §32.7(vi), §32.7 and Ch.32

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 ${\text{P}}_{\text{VI}}$ with

32.7.33		$z_1$	$=1/z_0,$
ⓘ Symbols: $z$: real Permalink: http://dlmf.nist.gov/32.7.E33 Encodings: TeX, pMML, png See also: Annotations for §32.7(vii), §32.7 and Ch.32
32.7.34		$z_2$	$=1-z_0,$
ⓘ 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		$z_3$	$=1/z_0,$
ⓘ 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		$$({\alpha }_1,{\beta }_1,{\gamma }_1,{\delta }_1)=({\alpha }_0,{\beta }_0,-{\delta }_0+\frac{1}{2},-{\gamma }_0+\frac{1}{2}),$$
ⓘ 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		${𝒮}_1:w_1(z_1)$	$=w_0(z_0)/z_0,$
ⓘ Symbols: $z$: real and ${𝒮}_j$: 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		${𝒮}_2:w_2(z_2)$	$=1-w_0(z_0),$
ⓘ Symbols: $z$: real and ${𝒮}_j$: 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		${𝒮}_3:w_3(z_3)$	$=1/w_0(z_0).$
ⓘ Symbols: $z$: real and ${𝒮}_j$: 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 ${𝒮}_j$, 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 ${\text{P}}_{\text{VI}}$ with

32.7.42		$$(\alpha ,\beta ,\gamma ,\delta )=(\frac{1}{2}{({\theta }_{\mathrm{\infty }}-1)}^2,-\frac{1}{2}{\theta }_0^2,\frac{1}{2}{\theta }_1^2,\frac{1}{2}(1-{\theta }_2^2)),$$
ⓘ 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 See also: Annotations for §32.7(vii), §32.7 and Ch.32

32.7.43		$$(A,B,C,D)=(\frac{1}{2}{({\mathrm{\Theta }}_{\mathrm{\infty }}-1)}^2,-\frac{1}{2}{\mathrm{\Theta }}_0^2,\frac{1}{2}{\mathrm{\Theta }}_1^2,\frac{1}{2}(1-{\mathrm{\Theta }}_2^2)),$$
ⓘ Symbols: ${\mathrm{\Theta }}_j$: 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		$${\theta }_j={\mathrm{\Theta }}_j+\frac{1}{2}\sigma ,$$
ⓘ Symbols: ${\mathrm{\Theta }}_j$: 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 $j=0,1,2,\mathrm{\infty }$, where

32.7.45		$$\sigma ={\theta }_0+{\theta }_1+{\theta }_2+{\theta }_{\mathrm{\infty }}-1=1-({\mathrm{\Theta }}_0+{\mathrm{\Theta }}_1+{\mathrm{\Theta }}_2+{\mathrm{\Theta }}_{\mathrm{\infty }}).$$
ⓘ Symbols: ${\mathrm{\Theta }}_j$: 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		$$\frac{\sigma }{w-W}=\frac{z(z-1)W^{\prime }}{W(W-1)(W-z)}+\frac{{\mathrm{\Theta }}_0}{W}+\frac{{\mathrm{\Theta }}_1}{W-1}+\frac{{\mathrm{\Theta }}_2-1}{W-z}=\frac{z(z-1)w^{\prime }}{w(w-1)(w-z)}+\frac{{\theta }_0}{w}+\frac{{\theta }_1}{w-1}+\frac{{\theta }_2-1}{w-z}.$$
ⓘ Symbols: $z$: real, ${\mathrm{\Theta }}_j$: transformation, ${\theta }_j$: parameters, $\sigma$ and $W(\zeta ,\epsilon /2)$: transformation Permalink: http://dlmf.nist.gov/32.7.E46 Encodings: TeX, pMML, png See also: Annotations for §32.7(vii), §32.7 and Ch.32

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

32.7.47		$u_1({\zeta }_1)$	$={\displaystyle \frac{(1-w)(w-z)}{{(1+\sqrt{z})}^{2}w}},$
		${\zeta }_1$	$={\left({\displaystyle \frac{1-\sqrt{z}}{1+\sqrt{z}}}\right)}^2,$
ⓘ 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 ${\text{P}}_{\text{VI}}$ with $\alpha =-\beta$ and $\gamma =\frac{1}{2}-\delta$ to ${\text{P}}_{\text{VI}}$ with $({\alpha }_1,{\beta }_1,{\gamma }_1,{\delta }_1)=(4\alpha ,-4\gamma ,0,\frac{1}{2})$. The quartic transformation

32.7.48		$u_2({\zeta }_2)$	$={\displaystyle \frac{{(w^2-z)}^2}{4w(w-1)(w-z)}},$
		${\zeta }_2$	$=z,$
ⓘ 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 See also: Annotations for §32.7(vii), §32.7 and Ch.32

transforms ${\text{P}}_{\text{VI}}$ with $\alpha =-\beta =\gamma =\frac{1}{2}-\delta$ to ${\text{P}}_{\text{VI}}$ with $({\alpha }_2,{\beta }_2,{\gamma }_2,{\delta }_2)=(16\alpha ,0,0,\frac{1}{2})$. 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 ${\text{P}}_{\text{VI}}$ with $\alpha =\beta =0$ and $\gamma =\frac{1}{2}-\delta$ to ${\text{P}}_{\text{VI}}$ with ${\alpha }_3={\beta }_3$ and ${\gamma }_3=\frac{1}{2}-{\delta }_3$.

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