Boys functions and lower incomplete-gamma notation

Write ${\mathbb{N}}_0=\{0,1,2,\dots \}$. For $m\in {\mathbb{N}}_0$ and $T\in \mathbb{R}$, define the finite-integral extension $F_m(T)={\int }_0^1{t}^{2m}\exp (-T{t}^2)dt\mathrm{.}$ The Boys function is the restriction to the conventional domain $T\ge 0$. The extension to negative $T$ is used only for ordinary center derivatives at the boundary value $T=0$. For $s>0$ and $T>0$, write the lower incomplete-gamma integral as $\gamma (s,T)={\int }_0^T{u}^{s-1}\exp (-u)du\mathrm{.}$ The complete Euler Gamma function $\mathrm{\Gamma }(s)$ has the existing meaning in Euler's Gamma function on the right half-plane, namely $\mathrm{\Gamma }(s)={\int }_0^{\mathrm{\infty }}{u}^{s-1}{e}^{-u}du$ for real $s>0$.

The integrand defining $F_m$ is continuous on $[0,1]$ and positive on $(0,1]$, so $F_m(T)$ exists, is positive, and, for $T\ge 0$, satisfies $F_m(T)\le 1\mathrm{/}(2m+1)$. The index $m$ is an arbitrary nonnegative integer; no fixed maximum order is part of this definition.