Convergence acceleration of the Gauss-Laguerre quadrature formula (Q1294544)

The author studies the approximation of integrals of the form \(\int_0^\infty x^\alpha e^{-x}f(x) dx\) for \(\alpha \geq 0\) by the Gauss-Laguerre quadrature formula. Assuming that \(f\) is analytic in \((0,\infty)\) with a singularity at 0 and an asymptotic expansion in terms of \(e^z z^\beta z^{\alpha_j}(\log z)^{\beta _j}\) as \(\text{ Re} z \rightarrow \infty\) with \(\beta >-1\), an asymptotic expansion of the error term of the quadrature formula is derived. Based on this expansion, extrapolation schemes can be applied to improve the numerical approximations to the integral. This is illustrated at four examples.