Gaussian Quadrature and Borwein Integrals

We discuss how to apply the Gaussian quadrature rule to evaluate the Borwein integral \[ \textrm{pr}_h(\Delta|\bar{c}) = \frac{1}{2\pi} \int_{-\infty}^{+\infty}\cos\Delta t\prod_{m=k+1}^{k+h} \frac{\sin \bar{c}Q^mt}{\bar{c}Q^mt}\,dt, \qquad k\in\mathbb{N}, \quad h\in\mathbb{N}_+,\quad \bar{c}, \Delta\in(0,\infty). \]

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