# Integral of a Gaussian multiplied with a Confluent Hypergeometric Function?

$$\int_0^{\infty}\!\!\mathrm{d}x~x^2\,e^{-\alpha x^2+i\beta x}\,_1F_1(a,2,icx)$$ Here, $\alpha,c>0$, $\beta\in\mathbb{R}$, $a\in\mathbb{C}$ and $_1F_1(\dotsi)$ is the Confluent hypergeometric function of the first kind [see, http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheFirstKind.html]. Can this integral be evaluated in closed-form?

• I had a long discussion about this form and the only way forward seemed to be going to the G function integral:functions.wolfram.com/HypergeometricFunctions/MeijerG/21/02/02 Having said that, the "2" in the parameters has always seemed a likely target for contiguous representation simplification. (get thee behind me satan!) I have not been able to exploit it though. – rrogers Jan 19 '18 at 14:19
• Thank for the suggestion, I will try the g-function way. – Siddhant Das Jan 19 '18 at 20:40