Hermite Gaussian and Boys Coulomb auxiliary functions

Using ${\mathbb{N}}_0=\{0,1,2,\dots \}$ from Cartesian Gaussian primitives, contractions, and integral conventions, for $p>0$, $P\in {\mathbb{R}}^3$, and $t,u,v\in {\mathbb{N}}_0$, the Hermite Gaussian is ${\mathrm{\Lambda }}_{tuv}(x;p,P)=\frac{{\mathrm{\partial }}^{t+u+v}}{\mathrm{\partial }{P}_1^t\mathrm{\partial }{P}_2^u\mathrm{\partial }{P}_3^v}\exp (-p\mathrm{\parallel }x-P{\mathrm{\parallel }}_2^2)\mathrm{.}$ For $\eta >0$, $P,C\in {\mathbb{R}}^3$, and $n,t,u,v\in {\mathbb{N}}_0$, define the Boys Coulomb auxiliary function $R_{tuv}^n(\eta ,P-C)=(-2\eta {)}^n\frac{{\mathrm{\partial }}^{t+u+v}}{\mathrm{\partial }{P}_1^t\mathrm{\partial }{P}_2^u\mathrm{\partial }{P}_3^v}{F}_n\text{\hspace{0.17em}⁣}(\eta \mathrm{\parallel }P-C{\mathrm{\parallel }}_2^2)\mathrm{.}$ Each derivative is an ordinary derivative of a smooth real function of the displayed center coordinates. For the Gaussian this follows from The exponential function is smooth and $(\exp {)}^{\mathrm{\prime }}=\exp$ and repeated coordinatewise product and chain rules in Sum, product, quotient, and chain rules from difference quotients. For the Boys composition, Boys integral, power series, recurrence, and incomplete-gamma representation gives $F_n^{(k)}(T)=(-1{)}^k{F}_{n+k}(T)$ by iteration on the full real extension; these derivatives are continuous by its integral representation and compact-uniform domination. The chain rules therefore also apply when $P=C$, where the Boys argument is zero. The multi-index $(t,u,v)$ records derivative order and $n$ records Boys order; both are nonnegative integers.