Consider the integral
$$f_k (x) := \int_{- \infty}^{+ \infty} dt \, t^k \, e^{-t^2 + x t} \, , \quad k \in \mathbb{N}.$$
According to Mathematica:
$$f_k(x) = \frac{1 -(-1)^k}{2} \, x \, \Gamma \left( 1+ \frac{k}{2} \right) \, \, _1 F_1 \left( 1+ \frac{k}{2}, \frac{3}{2}, \frac{x^2}{4} \right) \\ \qquad \qquad + \frac{1 +(-1)^k}{2} \, \Gamma \left( \frac{1+k}{2} \right) \, \, _1 F_1 \left( \frac{1+k}{2}, \frac{1}{2}, \frac{x^2}{4} \right) ,$$
in terms of the gamma function $\Gamma(x)$ and the hypergeometric function $_1F_1(x,y,z)$. At least for the first few terms, this can be written as
$$f_k(x) = \sqrt{\pi} \, e^{x^2/4} \, p_k(x/2) ,$$
where $p_k(x)$ are the polynomials
$$p_0(x) = 1\\ p_1(x) = x \\ p_2(x) = x^2 + \frac{1}{2} \\ p_3(x) = x^3 + \frac{3}{2} x \\ p_4(x) = x^4 + 3x^2 + \frac{3}{4}\\ p_5(x) = x^5 + 5x^3 + \frac{15}{4}x\\ ...$$
Are these known polynomials? Are they orthogonal wrt some weight function?