# §13.26 Addition and Multiplication Theorems

## §13.26(i) Addition Theorems for $\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(z\right)$

The function $\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(x+y\right)$ has the following expansions:

 13.26.1 $e^{-\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\mu-\frac{1}{2}}\sum_{n=0}^{% \infty}\frac{{\left(-2\mu\right)_{n}}}{n!}\left(\frac{-y}{\sqrt{x}}\right)^{n}% \*\mathop{M_{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}\/}\nolimits\!\left(x\right),$ $|y|<|x|$, 13.26.2 $e^{-\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\mu+\frac{1}{2}}\sum_{n=0}^{% \infty}\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{n}}}{{\left(1+2\mu\right)_{% n}}n!}\left(\frac{y}{\sqrt{x}}\right)^{n}\*\mathop{M_{\kappa-\frac{1}{2}n,\mu+% \frac{1}{2}n}\/}\nolimits\!\left(x\right),$ 13.26.3 $e^{-\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\kappa}\sum_{n=0}^{\infty}\frac{{% \left(\frac{1}{2}+\mu-\kappa\right)_{n}}y^{n}}{n!(x+y)^{n}}\mathop{M_{\kappa-n% ,\mu}\/}\nolimits\!\left(x\right),$ $\Re{(y/x)}>-\frac{1}{2}$, 13.26.4 $e^{\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\mu-\frac{1}{2}}\sum_{n=0}^{\infty% }\frac{{\left(-2\mu\right)_{n}}}{n!}\left(\frac{-y}{\sqrt{x}}\right)^{n}\*% \mathop{M_{\kappa+\frac{1}{2}n,\mu-\frac{1}{2}n}\/}\nolimits\!\left(x\right),$ $|y|<|x|$, 13.26.5 $e^{\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\mu+\frac{1}{2}}\sum_{n=0}^{\infty% }\frac{{\left(\frac{1}{2}+\mu+\kappa\right)_{n}}}{{\left(1+2\mu\right)_{n}}n!}% \left(\frac{-y}{\sqrt{x}}\right)^{n}\*\mathop{M_{\kappa+\frac{1}{2}n,\mu+\frac% {1}{2}n}\/}\nolimits\!\left(x\right),$ 13.26.6 $e^{\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\kappa}\sum_{n=0}^{\infty}\frac{{% \left(\frac{1}{2}+\mu+\kappa\right)_{n}}y^{n}}{n!(x+y)^{n}}\mathop{M_{\kappa+n% ,\mu}\/}\nolimits\!\left(x\right),$ $\Re{((y+x)/x)}>\frac{1}{2}$.

## §13.26(ii) Addition Theorems for $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)$

The function $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(x+y\right)$ has the following expansions:

 13.26.7 $e^{-\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\mu-\frac{1}{2}}\sum_{n=0}^{% \infty}\frac{{\left(\frac{1}{2}-\mu-\kappa\right)_{n}}}{n!}\left(\frac{-y}{% \sqrt{x}}\right)^{n}\*\mathop{W_{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}\/}% \nolimits\!\left(x\right),$ $|y|<|x|$, 13.26.8 $e^{-\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\mu+\frac{1}{2}}\sum_{n=0}^{% \infty}\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{n}}}{n!}\left(\frac{-y}{% \sqrt{x}}\right)^{n}\*\mathop{W_{\kappa-\frac{1}{2}n,\mu+\frac{1}{2}n}\/}% \nolimits\!\left(x\right),$ $|y|<|x|$, 13.26.9 $e^{-\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\kappa}\sum_{n=0}^{\infty}\frac{{% \left(\frac{1}{2}+\mu-\kappa\right)_{n}}{\left(\frac{1}{2}-\mu-\kappa\right)_{% n}}}{n!}\*\left(\frac{y}{x+y}\right)^{n}\mathop{W_{\kappa-n,\mu}\/}\nolimits\!% \left(x\right),$ $\Re{(y/x)}>-\frac{1}{2}$, 13.26.10 $e^{\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\mu-\frac{1}{2}}\sum_{n=0}^{\infty% }\frac{1}{n!}\left(\frac{-y}{\sqrt{x}}\right)^{n}\*\mathop{W_{\kappa+\frac{1}{% 2}n,\mu-\frac{1}{2}n}\/}\nolimits\!\left(x\right),$ $|y|<|x|$, 13.26.11 $e^{\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\mu+\frac{1}{2}}\sum_{n=0}^{\infty% }\frac{1}{n!}\left(\frac{-y}{\sqrt{x}}\right)^{n}\*\mathop{W_{\kappa+\frac{1}{% 2}n,\mu+\frac{1}{2}n}\/}\nolimits\!\left(x\right),$ $|y|<|x|$, 13.26.12 $e^{\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\kappa}\sum_{n=0}^{\infty}\frac{1}% {n!}\left(\frac{-y}{x+y}\right)^{n}\mathop{W_{\kappa+n,\mu}\/}\nolimits\!\left% (x\right),$ $\Re{(y/x)}>-\frac{1}{2}$.

## §13.26(iii) Multiplication Theorems for $\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(z\right)$ and $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)$

To obtain similar expansions for $\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(xy\right)$ and $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(xy\right)$, replace $y$ in the previous two subsections by $(y-1)x$.