= Scaled lower incomplete gamma integral
{title2=$F_a(x)=\int_0^1 e^{-xu}u^{a-1}\,du$}
For $a>0$, this <integral> is an <entire function> of $x$: the <exponential series> converges uniformly on bounded subsets after multiplication by the <integrable> bound $e^{|x|}u^{a-1}$. Thus
$$
F_a(x)=\sum_{m=0}^{\infty}\frac{(-x)^m}{m!(a+m)},\qquad F_a^{(m)}(0)=\frac{(-1)^m}{a+m}.
$$
For positive real $x$, a <change of variables> gives $F_a(x)=x^{-a}\gamma(a,x)$, where here $\gamma(a,x)=\int_0^xe^{-t}t^{a-1}dt$ denotes the lower <incomplete gamma function>. The singular-looking factor $x^{-a}$ is removed by the integral over $[0,1]$. Applying <integration by parts> to the <derivative> of $u^ae^{-xu}$ gives $xF_a'+aF_a=e^{-x}$; differentiating and eliminating $e^{-x}$ proves $xF_a''+(a+1+x)F_a'+aF_a=0$.
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