# §12.13 Sums

###### Contents

 12.13.1 $U\left(a,x+y\right)=e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\frac% {(-y)^{m}}{m!}U\left(a-m,x\right),$
 12.13.2 $U\left(a,x+y\right)=e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \genfrac{(}{)}{0.0pt}{}{-a-\tfrac{1}{2}}{m}y^{m}U\left(a+m,x\right),$
 12.13.3 $V\left(a,x+y\right)=e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \genfrac{(}{)}{0.0pt}{}{a-\tfrac{1}{2}}{m}y^{m}V\left(a-m,x\right),$
 12.13.4 $V\left(a,x+y\right)=e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \frac{y^{m}}{m!}V\left(a+m,x\right).$
 12.13.5 $U\left(a,x\cos t+y\sin t\right)\\ =e^{\frac{1}{4}(x\sin t-y\cos t)^{2}}\*\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt% }{}{-a-\tfrac{1}{2}}{m}(\tan t)^{m}U\left(m+a,x\right)U\left(-m-\tfrac{1}{2},y% \right),$ $\Re a\leq-\tfrac{1}{2},0\leq t\leq\tfrac{1}{4}\pi$.
 12.13.6 $n!U\left(n+\tfrac{1}{2},z\right)=i^{n}e^{-\frac{1}{2}z^{2}}\operatorname{erfc}% (z/\sqrt{2})U\left(-n-\tfrac{1}{2},iz\right)+\sum_{m=1}^{\left\lfloor\frac{1}{% 2}n+\frac{1}{2}\right\rfloor}U\left(2m-n-\tfrac{1}{2},z\right),$ $n=0,1,2,\dots.$
For $\operatorname{erfc}$ see §7.2(i).