Scaled lower incomplete gamma integral ($F_a(x)={\int }_0^1{e}^{-xu}{u}^{a-1}du$)

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 $e^{|x|}{u}^{a-1}$. ThusFor positive real $x$, a change of variables gives $F_a(x)=x^{-a}\gamma (a,x)$, where here $\gamma (a,x)={\int }_0^x{e}^{-t}{t}^{a-1}dt$ 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 $u^a{e}^{-xu}$ gives $x{F}_a^{\prime }+a{F}_a=e^{-x}$; differentiating and eliminating $e^{-x}$ proves $x{F}_a^{\prime \prime }+(a+1+x)F_a^{\prime }+a{F}_a=0$.