Modified discrete Laguerre polynomials for efficient computation of exponentially bounded Matsubara sums
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We develop a new type of orthogonal polynomial, the modified discrete Laguerre (MDL) polynomials, designed to accelerate the computation of bosonic Matsubara sums in statistical physics. The MDL polynomials lead to a rapidly convergent Gaussian "quadrature" scheme for Matsubara sums, and more generally for any sum F(0)/2 + F(h) + F(2h) + ⋯ of exponentially decaying summands F(nh) = f(nh)e^-nhs where hs>0. We demonstrate this technique for computation of finite-temperature Casimir forces arising from quantum field theory, where evaluation of the summand F requires expensive electromagnetic simulations. A key advantage of our scheme, compared to previous methods, is that the convergence rate is nearly independent of the spacing h (proportional to the thermodynamic temperature). We also prove convergence for any polynomially decaying F.
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