For the $n$-th order polynomial $p_n(x)$, I want to calculate this integral:
$$\int_0^1 p_n(x)e^{-a(x-b)^2}dx$$
I wasn't able to find any solutions. If a solution for this or a simplified variation of it already exists, I'd really appreciate it if you can direct me to it.
What I've tried:
One way I have tried is to unpack $p_n(x)$ and calculate:
$$f_k(x)=\int x^ke^{-a(x-b)^2}dx$$
Then, I tried to solve this using integration by parts. Let:
$$g_k(x)=x^ke^{-a(x-b)^2}$$ Then: $$\frac{dg_k(x)}{dx}=kg_{k-1}(x)-2ag_{k+1}(x)+abg_k(x)$$ Integrating on both sides, we get:
$$g_k(x)=kf_{k-1}(x)-2af_{k+1}(x)+abf_k(x)$$
Cleaning it up and dropping $(x)$ for readability:
$$f_k=\frac{b}{2}f_{k-1}+\frac{k-1}{2a}f_{k-2}-\frac{1}{2a}g_{k-1}$$
Now this is a nonhomogeneous recursive sequence with variable coefficients. Note, that $f_0$ and $f_1$ can be computed somewhat analytically based on the error function. To solve the sequence, I'm trying to apply this method. But, this method itself leads to messy integrations. Is there more straight-forward way to solve any of these steps?
Special Case
If $b=0$, we can simplify $f_k$ to the lower incomplete gamma function:
$$\int_0^1x^ke^{-ax^2}dx=\frac{1}{2}\sqrt{a}^{k+1}\gamma(\frac{k+1}{2},a)$$
I wonder if gamma function would be part of the general answer as well.
Thank you.
