## §19.26(i) General Formulas

In this subsection, and also §§19.26(ii) and 19.26(iii), we assume that $\lambda,x,y,z$ are positive, except that at most one of $x,y,z$ can be 0.

 19.26.1 $\mathop{R_{F}\/}\nolimits\!\left(x+\lambda,y+\lambda,z+\lambda\right)+\mathop{% R_{F}\/}\nolimits\!\left(x+\mu,y+\mu,z+\mu\right)=\mathop{R_{F}\/}\nolimits\!% \left(x,y,z\right),$

where $\mu>0$ and

 19.26.2 $x+\mu=\lambda^{-2}\left(\sqrt{(x+\lambda)yz}+\sqrt{x(y+\lambda)(z+\lambda)}% \right)^{2},$ Symbols: $\lambda$: positive, $x$: positive, $y$: positive, $z$: positive and $\mu>0$: parameter Referenced by: §19.26(i) Permalink: http://dlmf.nist.gov/19.26.E2 Encodings: TeX, pMML, png See also: Annotations for 19.26(i)

with corresponding equations for $y+\mu$ and $z+\mu$ obtained by permuting $x,y,z$. Also,

 19.26.3 $\sqrt{z}=\frac{\xi\zeta^{\prime}+\eta^{\prime}\zeta-\xi\eta^{\prime}}{\sqrt{% \xi\eta\zeta^{\prime}}+\sqrt{\xi^{\prime}\eta^{\prime}\zeta}},$ Symbols: $z$: positive, $\xi$: coordinate, $\eta$: coordinate and $\zeta$: coordinate Permalink: http://dlmf.nist.gov/19.26.E3 Encodings: TeX, pMML, png See also: Annotations for 19.26(i)

where

 19.26.4 $\displaystyle(\xi,\eta,\zeta)$ $\displaystyle=(x+\lambda,y+\lambda,z+\lambda),$ $\displaystyle(\xi^{\prime},\eta^{\prime},\zeta^{\prime})$ $\displaystyle=(x+\mu,y+\mu,z+\mu),$

with $\sqrt{x}$ and $\sqrt{y}$ obtained by permuting $x$, $y$, and $z$. (Note that $\xi\zeta^{\prime}+\eta^{\prime}\zeta-\xi\eta^{\prime}=\xi^{\prime}\zeta+\eta% \zeta^{\prime}-\xi^{\prime}\eta$.) Equivalent forms of (19.26.2) are given by

 19.26.5 $\mu=\lambda^{-2}\left(\sqrt{xyz}+\sqrt{(x+\lambda)(y+\lambda)(z+\lambda)}% \right)^{2}-\lambda-x-y-z,$ Symbols: $\lambda$: positive, $x$: positive, $y$: positive, $z$: positive and $\mu>0$: parameter Permalink: http://dlmf.nist.gov/19.26.E5 Encodings: TeX, pMML, png See also: Annotations for 19.26(i)

and

 19.26.6 $(\lambda\mu-xy-xz-yz)^{2}=4xyz(\lambda+\mu+x+y+z).$ Symbols: $\lambda$: positive, $x$: positive, $y$: positive, $z$: positive and $\mu>0$: parameter Permalink: http://dlmf.nist.gov/19.26.E6 Encodings: TeX, pMML, png See also: Annotations for 19.26(i)

Also,

 19.26.7 $\mathop{R_{D}\/}\nolimits\!\left(x+\lambda,y+\lambda,z+\lambda\right)+\mathop{% R_{D}\/}\nolimits\!\left(x+\mu,y+\mu,z+\mu\right)=\mathop{R_{D}\/}\nolimits\!% \left(x,y,z\right)-\frac{3}{\sqrt{z(z+\lambda)(z+\mu)}},$
 19.26.8 $2\!\mathop{R_{G}\/}\nolimits\!\left(x+\lambda,y+\lambda,z+\lambda\right)+2\!% \mathop{R_{G}\/}\nolimits\!\left(x+\mu,y+\mu,z+\mu\right)=2\!\mathop{R_{G}\/}% \nolimits\!\left(x,y,z\right)+\lambda\mathop{R_{F}\/}\nolimits\!\left(x+% \lambda,y+\lambda,z+\lambda\right)+\mu\mathop{R_{F}\/}\nolimits\!\left(x+\mu,y% +\mu,z+\mu\right)+\sqrt{\lambda+\mu+x+y+z}.$
 19.26.9 $\mathop{R_{J}\/}\nolimits\!\left(x+\lambda,y+\lambda,z+\lambda,p+\lambda\right% )+\mathop{R_{J}\/}\nolimits\!\left(x+\mu,y+\mu,z+\mu,p+\mu\right)=\mathop{R_{J% }\/}\nolimits\!\left(x,y,z,p\right)-3\!\mathop{R_{C}\/}\nolimits\!\left(\gamma% -\delta,\gamma\right),$

where

 19.26.10 $\displaystyle\gamma$ $\displaystyle=p(p+\lambda)(p+\mu),$ $\displaystyle\delta$ $\displaystyle=(p-x)(p-y)(p-z).$

Lastly,

 19.26.11 $\mathop{R_{C}\/}\nolimits\!\left(x+\lambda,y+\lambda\right)+\mathop{R_{C}\/}% \nolimits\!\left(x+\mu,y+\mu\right)=\mathop{R_{C}\/}\nolimits\!\left(x,y\right),$

where $\lambda>0$, $y>0$, $x\geq 0$, and

 19.26.12 $\displaystyle x+\mu$ $\displaystyle=\lambda^{-2}(\sqrt{x+\lambda}y+\sqrt{x}(y+\lambda))^{2},$ $\displaystyle y+\mu$ $\displaystyle=(y(y+\lambda)/\lambda^{2})(\sqrt{x}+\sqrt{x+\lambda})^{2}.$ Symbols: $\lambda$: positive, $x$: positive, $y$: positive and $\mu>0$: parameter Permalink: http://dlmf.nist.gov/19.26.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 19.26(i)

Equivalent forms of (19.26.11) are given by

 19.26.13 $\mathop{R_{C}\/}\nolimits\!\left(\alpha^{2},\alpha^{2}-\theta\right)+\mathop{R% _{C}\/}\nolimits\!\left(\beta^{2},\beta^{2}-\theta\right)=\mathop{R_{C}\/}% \nolimits\!\left(\sigma^{2},\sigma^{2}-\theta\right),$ $\sigma=(\alpha\beta+\theta)/(\alpha+\beta)$, Symbols: $\mathop{R_{C}\/}\nolimits\!\left(\NVar{x},\NVar{y}\right)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $\alpha^{2}$: real or complex parameter Referenced by: §19.26(i), §19.26(ii) Permalink: http://dlmf.nist.gov/19.26.E13 Encodings: TeX, pMML, png See also: Annotations for 19.26(i)

where $0<\gamma^{2}-\theta<\gamma^{2}$ for $\gamma=\alpha,\beta,\sigma$, except that $\sigma^{2}-\theta$ can be 0, and

 19.26.14 $(p-y)\mathop{R_{C}\/}\nolimits\!\left(x,p\right)+(q-y)\mathop{R_{C}\/}% \nolimits\!\left(x,q\right)=(\eta-\xi)\mathop{R_{C}\/}\nolimits\!\left(\xi,% \eta\right),$ $x\geq 0$, $y\geq 0$; $p,q\in\mathbb{R}\setminus\{0\}$,

where

 19.26.15 $\displaystyle(p-x)(q-x)$ $\displaystyle=(y-x)^{2},$ $\displaystyle\xi$ $\displaystyle=y^{2}/x,$ $\displaystyle\eta$ $\displaystyle=pq/x,$ $\displaystyle\eta-\xi$ $\displaystyle=p+q-2y.$ Symbols: $x$: positive, $y$: positive, $\xi$: coordinate and $\eta$: coordinate Permalink: http://dlmf.nist.gov/19.26.E15 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for 19.26(i)

## §19.26(ii) Case $x=0$

If $x=0$, then $\lambda\mu=yz$. For example,

 19.26.16 $\mathop{R_{F}\/}\nolimits\!\left(\lambda,y+\lambda,z+\lambda\right)={\mathop{R% _{F}\/}\nolimits\!\left(0,y,z\right)-\mathop{R_{F}\/}\nolimits\!\left(\mu,y+% \mu,z+\mu\right),}$ $\lambda\mu=yz$.

An equivalent version for $\mathop{R_{C}\/}\nolimits$ is

 19.26.17 $\sqrt{\alpha}\mathop{R_{C}\/}\nolimits\!\left(\beta,\alpha+\beta\right)+\sqrt{% \beta}\mathop{R_{C}\/}\nolimits\!\left(\alpha,\alpha+\beta\right)=\pi/2,$ $\alpha,\beta\in\mathbb{C}\setminus(-\infty,0)$, $\alpha+\beta>0$.

## §19.26(iii) Duplication Formulas

 19.26.18 $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)=2\!\mathop{R_{F}\/}\nolimits\!% \left(x+\lambda,y+\lambda,z+\lambda\right)=\mathop{R_{F}\/}\nolimits\!\left(% \frac{x+\lambda}{4},\frac{y+\lambda}{4},\frac{z+\lambda}{4}\right),$

where

 19.26.19 $\lambda=\sqrt{x}\sqrt{y}+\sqrt{y}\sqrt{z}+\sqrt{z}\sqrt{x}.$ Symbols: $\lambda$: positive, $x$: positive, $y$: positive and $z$: positive Referenced by: §19.26(iii) Permalink: http://dlmf.nist.gov/19.26.E19 Encodings: TeX, pMML, png See also: Annotations for 19.26(iii)
 19.26.20 $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)=2\!\mathop{R_{D}\/}\nolimits\!% \left(x+\lambda,y+\lambda,z+\lambda\right)+\frac{3}{\sqrt{z}(z+\lambda)}.$
 19.26.21 $2\!\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)=4\!\mathop{R_{G}\/}\nolimits% \!\left(x+\lambda,y+\lambda,z+\lambda\right)-\lambda\mathop{R_{F}\/}\nolimits% \!\left(x,y,z\right)-\sqrt{x}-\sqrt{y}-\sqrt{z}.$
 19.26.22 $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)=2\!\mathop{R_{J}\/}\nolimits\!% \left(x+\lambda,y+\lambda,z+\lambda,p+\lambda\right)+3\!\mathop{R_{C}\/}% \nolimits\!\left(\alpha^{2},\beta^{2}\right),$

where

 19.26.23 $\displaystyle\alpha$ $\displaystyle=p(\sqrt{x}+\sqrt{y}+\sqrt{z})+\sqrt{x}\sqrt{y}\sqrt{z},$ $\displaystyle\beta$ $\displaystyle=\sqrt{p}(p+\lambda),$ $\displaystyle\beta\pm\alpha$ $\displaystyle=(\sqrt{p}\pm\sqrt{x})(\sqrt{p}\pm\sqrt{y})(\sqrt{p}\pm\sqrt{z}),$ $\displaystyle\beta^{2}-\alpha^{2}$ $\displaystyle=(p-x)(p-y)(p-z),$ Symbols: $\lambda$: positive, $\alpha$, $\beta$, $x$: positive, $y$: positive and $z$: positive Permalink: http://dlmf.nist.gov/19.26.E23 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for 19.26(iii)

either upper or lower signs being taken throughout.

The equations inverse to $z+\lambda=(\sqrt{z}+\sqrt{x})(\sqrt{z}+\sqrt{y})$ and the two other equations obtained by permuting $x,y,z$ (see (19.26.19)) are

 19.26.24 $z=(\xi\zeta+\eta\zeta-\xi\eta)^{2}/(4\xi\eta\zeta),$ $(\xi,\eta,\zeta)=(x+\lambda,y+\lambda,z+\lambda)$,

and two similar equations obtained by exchanging $z$ with $x$ (and $\zeta$ with $\xi$), or $z$ with $y$ (and $\zeta$ with $\eta$).

Next,

 19.26.25 $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)=2\!\mathop{R_{C}\/}\nolimits\!% \left(x+\lambda,y+\lambda\right),$ $\lambda=y+2\sqrt{x}\sqrt{y}$.

Equivalent forms are given by (19.22.22). Also,

 19.26.26 $\mathop{R_{C}\/}\nolimits\!\left(x^{2},y^{2}\right)=\mathop{R_{C}\/}\nolimits% \!\left(a^{2},ay\right),$ $a=(x+y)/2$, $\Re{x}\geq 0$, $\Re{y}>0$,

and

 19.26.27 $\mathop{R_{C}\/}\nolimits\!\left(x^{2},x^{2}-\theta\right)=2\!\mathop{R_{C}\/}% \nolimits\!\left(s^{2},s^{2}-\theta\right),$ $s=x+\sqrt{x^{2}-\theta}$, $\theta\neq x^{2}$ or $s^{2}$. Symbols: $\mathop{R_{C}\/}\nolimits\!\left(\NVar{x},\NVar{y}\right)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $x$: positive Permalink: http://dlmf.nist.gov/19.26.E27 Encodings: TeX, pMML, png See also: Annotations for 19.26(iii)