Error in Gauss-Laguerre numerical quadrature scheme The $n$-th Gauss-Laguerre quadrature scheme aims to approximate integral of exponentially decreassing function over $[0 ; \infty[$ by a finite sum, according to:
$$ \int _0
        ^{+ \infty}
        e^{-x} f(x)
        \ dx
   \approx
   \sum _{i = 1}
        ^n
        \omega_i f(x_i)
   \ ,
$$
where $x_1$, $\cdots$, $x_n$ are the roots of the $n$-th Laguerre polynomial $L_n$
and the weights $\omega_1$, $\cdots$, $\omega_n$ are chosen according to
$\omega_i = \dfrac{1}{x_i \big ( L_n'(x_i) \big )^2}$, $1 \leq i \leq n$.
$\bullet$ Let us denotes by $E_n(f)$ the error of the quadrature:
$E_n(f) = \displaystyle {  \int _0
                                ^{+ \infty}
                                e^{-x} f(x)
                                \ dx
                        }
           -
           \displaystyle {  \sum _{i = 1}
                                 ^n
                                 \omega_i f(x_i)
                         }
$.
The general estimation of the error of gaussian quadratures, specialized in Gauss-Laguerre scheme, is the following:

For all $n \in \mathbb{N}$, there exists $\xi \in ]0 ; \infty[$ such that
$E_n(f) = \dfrac{n!^2}{(2n)!} f^{(2n)}(\xi)$.

Unfortunately, this is unusable in many case, since we know nothing on this $\xi$. Therefore, we shall consider functions $f$ with derivatives satisfyng
$||f^{2n}||_{\infty, \mathbb{R}^+} = \mathcal{o}\left ( \dfrac{1}{n!^2} \right )$... This is a bit restrictive...
$\bullet$ We also know the Uspensky theorem:

$E_n(f) \underset{n \longrightarrow + \infty}{\longrightarrow} 0$ for functions $f$ satisfying $|f(x)| \leq c \dfrac{e^x}{x^{1 + \rho}}$ for large $x >> 1$, where some $\rho > 0$.

(See J. V. Uspenksy, On the convergence of quadrature formulas related to an infinite interval, Trans. Amer. Math. Soc. 30 (1928), 542-559)
But, this do not gives explicit result on the convergence speed to $0$, nor a upper bound of $E_n(f)$ going to $0$.
$\bullet$ So, what is the most precise result about the error term? In particular, how can we know which degree $n$ should we use to find a numerical approximation of $\displaystyle { \int _0 ^{+ \infty} e^{-x + \sqrt{x}} \ dx}$.
 A: The function $f(x)=e^{-x+\sqrt{x}}$ belongs to the space $C_{0}^{3}[0,\infty)$ defined, for $q\geq p\geq0$, by
$$C _ { p } ^ { q } [ 0,\infty ) : = \{ f \in C ^ { p } [ 0,\infty ) \cap C ^ { q } ( 0,\infty ),~x ^ { i } f ^ { ( p + i ) } ( x ) \in C [ 0,\infty ),~i=1,\ldots,q-p\}.$$
According to the result given in [1], the error rate behaves like
\begin{equation}\label{rate-quad}
\mathcal{O}(n^{-1})E_{n-1}(\Phi^{(3)}(x),e^{-x/2})=\mathcal{O}(n^{-1})E_{n-1}(\Phi^{(3)}(2x),e^{-x}),
\end{equation}
where $ \Phi ( x ) : = x ^ { 3} f ( x )=x^{3}e^{-x+\sqrt{x}}$, and $E _ { n } ( f ; w )$ is the rate of weighted polynomial approximation,
$$E _ { n } ( f ; w ) : = \inf_ { p _ { n } } \| w (f - p _ { n })\| _ { \infty ,[ 0,\infty ) }.$$
Here, $\Phi^{(3)}(x)=P_{6}(\sqrt{x})e^{-x+\sqrt{x}}$, where $P_{6}$ is a polynomial of degree
$6$.
Next, the rate of approximation $E_{n}(f,w)$, $w(x)=e^{-x}$, is known from [2] p.112, available here, namely
\begin{equation}\label{rate-En}
E_{n}(f,e^{-x})\leq C\frac{\sqrt{a_{n}}}{n}\left\|\sqrt{x}f'(x)e^{-x}\right\|_{\infty},
\end{equation}
for $f\in W_{1}^{\infty}(e^{-x})$, where $a_{n}\sim n$ is the so-called Mhaskar-Rakhmanov-Saff number, and, for $r\geq1$,
$W_{r}^{\infty}(e^{-x})$ is the Sobolev-type space,
$$
W_{r}^{\infty}(e^{-x})=\left\{f \in L_{e^{-x}}^{\infty}, f^{(r-1)}\text{ abs. continuous on }(0,\infty)\text { and }\left\|f^{(r)}(x) x^{r/2} e^{-x}\right\|_{\infty}<\infty\right\},
$$
$$
L_{e^{-x}}^{\infty}=\{f\in C((0,\infty)),~\lim_{x\to0,~x\to\infty} f(x)e^{-x}=0\}.
$$
Note that the function $f(x)=e^{-2x+\sqrt{2x}}$ belongs to $W_{1}^{\infty}(e^{-x})$, but not to any of the more regular spaces
$W_{r}^{\infty}(e^{-x})$, $r\geq2$ (for which the rate of approximation is improved to $(\sqrt{a_{n}}/n)^{r}$).
Finally, taking into account the two previous estimates, we get $\mathcal{O}(n^{-3/2})$ for the error rate of the Gauss-Laguerre quadrature for $f(x)=e^{-x+\sqrt{x}}$.
[1] G. Mastroianni, G. Monegato, Convergence of product integration rules over
$(0,\infty)$ for functions with weak singularities at the origin. Math. Comp. 64, (1995), 237--249.
[2] G. Mastroianni, J. Szabados,
Polynomial approximation on the real semiaxis with generalized Laguerre weights.
Stud. Univ. Babes-Bolyai Math. 52 (2007), nr.4, 105--128.
