Can $f(x)=\int_{0}^{\infty}t^{x^2+1}e^{-2t}dt$ be written as a power series in a neighbourhood of zero?

Question

Let $f(x)=\int_{0}^{\infty}t^{x^2+1}e^{-2t}dt$. Can $f(x)$ be written as a power series in a neighbourhood of zero? In this case, what would its convergence radius be?

Trying to solve this problem I have realized that the function f is similar to Euler's Gamma function, which can be written as a series of powers, so I think the answer should be yes. However I am not able to rigorously prove it.

Answer 1

The answer is yes. In particular, note that with a substitution we can rewrite $$ f(x) = 2^{-(x^2 + 1)} \int_0^\infty t^{x^2 + 1}e^{-t}\,dt = 2^{-(x^2 + 1)}\Gamma(x^2 + 2). $$