§19.26 Addition Theorems

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.

where $\mu >0$ and

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

where

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

and

Also,

where

Lastly,

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

Equivalent forms of (19.26.11) are given by

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

where

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

An equivalent version for $R_C$ is

where

where

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

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

Next,

Equivalent forms are given by (19.22.22). Also,

and