# §12.13 Sums

###### Contents

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