Exponential decay bound on integral I have an integral of the form
$$ \int_R^{\infty} e^{-x} x^n \vert L_m^{\alpha}(x) \vert^2 \ dx,$$
where $L_m^{\alpha}$ is the generalized Laguerre polynomial and $n \ge 0.$
I would to get a nice explicit exponential bound on this integral in terms of $R$ (I intend to take $R$ large). However, most upper bounds on these polynomials that I could find involve exponentials themselves, see for instance here on DLMF.
Does anybody see a way to get a nice estimate on this integral for $R$ large? The exponential decay is clear, as we are integrating an exponential against a polynomial. However, I would like to have a bound that is a good tradeoff between compact and efficient.
 A: Let $a:=\alpha$. The integral in question is
\begin{equation*}
    I(R):=\int_R^\infty e^{-x} x^n L_m^a(x)^2\, dx, 
\end{equation*}
where
\begin{equation*}
    L_m^a(x)=\sum_{i=0}^m b_i,\quad b_i:=(-1)^i \binom{m+a}{m-i}\frac{x^i}{i!},
\end{equation*}
so that for $i=0,\dots,m-1$ and $x\ge R>0$
\begin{equation*}
    \frac{|b_i|}{|b_{i+1}|}=\frac{i+1}{(m-i)x}\,|i+1+a|\le\frac{c_{m,a}}R,\quad 
    c_{m,a}:=m\max(|m+a|,|a+1|)
\end{equation*}
and hence for $R>c_{m,a}$
\begin{equation*}
    L_m^a(x)^2\le\frac{b_m^2}{(1-c_{m,a}/R)^2}
    =\frac1{(1-c_{m,a}/R)^2}\frac{x^{2m}}{(m!)^2}. 
\end{equation*}
Thus, for $R>\max(c_{m,a},n+2m)$, by Proposition 2.7,
\begin{equation*}
\begin{aligned}
    I(R)&\le J(R):=\frac1{(1-c_{m,a}/R)^2}\frac1{(m!)^2}\,\frac{e^{-R}R^{n+2m}}{1-(n+2m)/R} \\ 
    &\sim\frac{e^{-R}R^{n+2m}}{(m!)^2} 
\end{aligned}
\tag{1}\label{1}
\end{equation*}
as $R\to\infty$. So, we do get an exponential decrease in $R$.
The upper bound $J(R)$ on $I(R)$ in \eqref{1} is asymptotically exact (as $R\to\infty$), since $L_m^a(x)^2\sim\dfrac{x^{2m}}{(m!)^2}$ as $x\to\infty$.
