Closed Form of Infinite Series Used in Some Atomic Integrals Containing r12, r13, and r23

In an excellent scheme recently developed for evaluating the integral $${\displaystyle \int }{f}_1{\mathrm{(r}}_1{\mathrm{)f}}_2{\mathrm{(r}}_2{\mathrm{)f}}_3{\mathrm{(r}}_3{\mathrm{)g}}_1{\mathrm{(r}}_{12}{\mathrm{)g}}_2{\mathrm{(r}}_{23}{\mathrm{)g}}_3{\mathrm{(r}}_{13}\mathrm{)(dv)}$$ met in the calculation of correlated atomic wavefunctions, certain functions required in the computational scheme had to be evaluated by an infinite series expansion. As many as 40 terms may be needed in each of the three required infinite summations to get eight significant figures. We give a closed form expression for such functions avoiding all infinite sums. The new result is very compact and avoids the previous difficulty of numerical stability.