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Let $I(x,a)=\int_a^\infty\frac{x^t}{\Gamma(t+1)}dt$. $I(x,a)$ converges for $x\in\mathbb{R}^+, a\in\mathbb{R}$. Note $\frac{\partial}{\partial x}I(x,a)=I(x,a-1)=I(x,a)+\int_{a-1}^a\frac{x^t}{\Gamma(t+1)}dt$ This integral was inspired by generalizing the Taylor series of $\exp(x)$ to a more continuous case.

I sincerely hope that someone has thought of this integral in more detail and will share further insight.

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