Integration of Gaussian $\times$ Laguerre functions

Question

What is $\begin{align} \int_{-\infty}^{\infty}{e^{-ax^2+bx+c}L_n(dx^2+ex+f)dx} \end{align}$ ?

Are there any identities that are close to this form that could be helpful?

Answer 1

Better to concentrate by an affine change of variable to $$u_n=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-x^2/2}L_n(P(x))dx=\mathbb{E}(L_n(P(X)))$$ where $X\sim N(0,1)$ and $P$ is a second degree polynomial. Using $$ \frac{1}{1-t}e^{-\frac{xt}{1-t}}=\sum _{n=0}^{\infty}\frac{L_n(x)}{n!}t^n$$ you get a generating function of $(u_n)$ by the computable Gaussian integral $$\sum _{n=0}^{\infty}\frac{u_n}{n!}t^n=\frac{1}{1-t}\mathbb{E}\left(\exp(-\frac{P(X)t}{1-t})\right).$$

Answer 2

$$\frac{1}{1-t}\int_{-\infty}^{\infty}e^{-\frac{z^2}{2}+(az^2+bz+c)\frac{1}{1-t}}\frac{dz}{\sqrt{2\pi}}=\frac{1}{1-t}\times e^{\frac{s}{1-t}}\times e^{\frac{s_1}{1-t(1+a)}}$$ where $s$ and $s_1$ are not too complicated constants depending on $a,b,c.$ Define $P_n(s)$ by $$\sum_{n=0}^{\infty}P_n(s)t^n=e^{\frac{s}{1-t}}$$ This polynomial of degree $n$ is important since $$\sum_{n=0}^{\infty}P_n(s_1)(1+a)^nt^n=e^{\frac{s_1}{1-t(1+a)}}$$ A way to compute $P_n(s)$ by induction is by introducing $$P_n(s,t)=e^{-\frac{s}{1-t}}\frac{1}{n!}\left(\frac{d}{dt}\right)^ne^{\frac{s}{1-t}}$$ satisfying $P_n(s)=P_n(s,0)$ as well as the induction formula $$P_{n+1}(s,t)=\frac{1}{n+1}\left(\frac{s}{(1-t)^2}P_{n}(s,t)+\frac{\partial}{\partial t}P_{n}(s,t)\right).$$

Answer 3

Laguerre polynomials and associated Laguerre polynomials written as parametric differentiation/derivatives with two parameters that may be useful in this context:

$L_n(-z_1z_2)=\frac{1}{n!}\lim_{A\rightarrow 0}\frac{\partial^n}{\partial A^n}\lim_{B\rightarrow0}\frac{\partial^n}{\partial B^n}\left(\exp\left(AB+Az_1+Bz_2\right)\right)$

and

$L_n^\alpha(-z_1z_2)=\frac{z_2^{-\alpha}}{n!}\lim_{A\rightarrow0}\frac{\partial^n}{\partial A^n}\lim_{B\rightarrow0}\frac{\partial^{(n+a)}}{\partial B^{(n+a)}}\left(\exp\left(AB+Az_1+Bz_2\right)\right)$