Asymptotic expansions of Gauss-Legendre quadrature rules for integrals with endpoint singularities

Let $I[f]=\int_{-1}^1f(x)\,dx,$ where $f\in C^\infty(-1,1)$, and let $G_n[f]=\sum^n_{i=1}w_{ni}f(x_{ni})$ be the $n$-point Gauss-Legendre quadrature approximation to $I[f]$. In this paper, we derive an asymptotic expansion as $n\to\infty$ for the error $E_n[f]=I[f]-G_n[f]$ when $f(x)$ has general algebraic-logarithmic singularities at one or both endpoints. We assume that $f(x)$ has asymptotic expansions of the forms

$f(x)$	$\sim\sum^\infty_{s=0}U_s(\log (1-x))(1-x)^{\alpha_s}$   as $x\to 1-$,
$f(x)$	$\sim\sum^\infty_{s=0}V_s(\log(1+x))(1+x)^{\beta_s}$   as $x\to -1+$,

where $U_s(y)$ and $V_s(y)$ are some polynomials in $y$. Here, $\alpha_s$ and $\beta_s$ are, in general, complex and $\Re\alpha_s,\Re\beta_s>-1$. An important special case is that in which $U_s(y)$ and $V_s(y)$ are constant polynomials; for this case, the asymptotic expansion of $E_n[f]$ assumes the form

$$E_n[f]\sim\sum^\infty_{\substack{s=0 \alpha_s\not\in \mathbb{Z... ... \mathbb{Z}^+}}\sum^\infty_{i=1}b_{si}h^{\beta_s+i}\quad\text{as $n\to\infty$},$$

where $h=(n+1/2)^{-2}$, $\mathbb{Z}^+=\{0,1,2,\ldots\},$ and $a_{si}$ and $b_{si}$ are constants independent of $n$.