# §13.13 Addition and Multiplication Theorems

## §13.13(i) Addition Theorems for $M\left(a,b,z\right)$

The function $M\left(a,b,x+y\right)$ has the following expansions:

 13.13.1 $\sum_{n=0}^{\infty}\frac{{\left(a\right)_{n}}y^{n}}{{\left(b\right)_{n}}n!}M% \left(a+n,b+n,x\right),$
 13.13.2 $\left(\frac{x+y}{x}\right)^{1-b}\sum_{n=0}^{\infty}\frac{{\left(1-b\right)_{n}% }(-\ifrac{y}{x})^{n}}{n!}M\left(a,b-n,x\right),$ $|y|<|x|$,
 13.13.3 $\left(\frac{x}{x+y}\right)^{a}\sum_{n=0}^{\infty}\frac{{\left(a\right)_{n}}y^{% n}}{n!(x+y)^{n}}M\left(a+n,b,x\right),$ $\Re(y/x)>-\tfrac{1}{2}$,
 13.13.4 $e^{y}\sum_{n=0}^{\infty}\frac{{\left(b-a\right)_{n}}(-y)^{n}}{{\left(b\right)_% {n}}n!}M\left(a,b+n,x\right),$
 13.13.5 $e^{y}\left(\frac{x}{x+y}\right)^{b-a}\sum_{n=0}^{\infty}\frac{{\left(b-a\right% )_{n}}y^{n}}{n!(x+y)^{n}}\*M\left(a-n,b,x\right),$ $\Re((y+x)/x)>\frac{1}{2}$,
 13.13.6 $e^{y}\left(\frac{x+y}{x}\right)^{1-b}\sum_{n=0}^{\infty}\frac{{\left(1-b\right% )_{n}}(-y)^{n}}{n!x^{n}}\*M\left(a-n,b-n,x\right),$ $|y|<|x|$.

## §13.13(ii) Addition Theorems for $U\left(a,b,z\right)$

The function $U\left(a,b,x+y\right)$ has the following expansions:

 13.13.7 $\sum_{n=0}^{\infty}\frac{{\left(a\right)_{n}}(-y)^{n}}{n!}U\left(a+n,b+n,x% \right),$ $|y|<|x|$,
 13.13.8 $\left(\frac{x+y}{x}\right)^{1-b}\*\sum_{n=0}^{\infty}\frac{{\left(1+a-b\right)% _{n}}(-\ifrac{y}{x})^{n}}{n!}U\left(a,b-n,x\right),$ $|y|<|x|$,
 13.13.9 $\left(\frac{x}{x+y}\right)^{a}\sum_{n=0}^{\infty}\frac{{\left(a\right)_{n}}{% \left(1+a-b\right)_{n}}y^{n}}{n!(x+y)^{n}}U\left(a+n,b,x\right),$ $\Re(y/x)>-\tfrac{1}{2}$,
 13.13.10 $e^{y}\sum_{n=0}^{\infty}\frac{(-y)^{n}}{n!}U\left(a,b+n,x\right),$ $|y|<|x|$,
 13.13.11 $e^{y}\left(\frac{x}{x+y}\right)^{b-a}\sum_{n=0}^{\infty}\frac{(-y)^{n}}{n!(x+y% )^{n}}U\left(a-n,b,x\right),$ $\Re(y/x)>-\tfrac{1}{2}$,
 13.13.12 $e^{y}\left(\frac{x+y}{x}\right)^{1-b}\sum_{n=0}^{\infty}\frac{(-y)^{n}}{n!x^{n% }}U\left(a-n,b-n,x\right),$ $|y|<|x|$.

## §13.13(iii) Multiplication Theorems for $M\left(a,b,z\right)$ and $U\left(a,b,z\right)$

To obtain similar expansions for $M\left(a,b,xy\right)$ and $U\left(a,b,xy\right)$, replace $y$ in the previous two subsections by $(y-1)x$.