I want to evaluate the improper integral of $\int_{0}^{\infty} e^{(ax)} U(c,d,b_1x)U(c,d,b_2x) \, dx$ where U is the confluent hypergeometric function, $d$ is a real number and $d \in (1,2)$, $a$ and $c$ are complex numbers with both real parts $\Re(a) >0$ and $ \Re(b_1), \Re(b_2) < 0 $
Here is my attempt:
- For starters, I tried to evaluate $\int_{0}^{\infty} e^{(x)} U(1,1,x) \, dx$ (a simple case) in Wolfram, there seems to be closed form solution. The indefinite integral and plots for various values of $(a,b)$ are here. However, I am unable to compute $\int_{0}^{\infty} e^{(x)} U(1,1,x) U(1,1,x) \, dx$ even with Wolfram.
If I can understand the step-by-step process of evaluating the integral for the simple case, I should be able to evaluate even for the general case, but unfortunately, Wolfram does not provide it.
I would be thankful if someone can help me with this integration of $$\int_{0}^{\infty} e^{(ax)} U(c,d,b_1x)U(c,d,b_2x) \, dx$$