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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Gaussian quadrature involving Einstein and Fermi functions with an application to summation of series


Authors: Walter Gautschi and Gradimir V. Milovanović
Journal: Math. Comp. 44 (1985), 177-190, S1
MSC: Primary 65D32; Secondary 33A65, 65A05, 65B10, 81-08, 82-08
MathSciNet review: 771039
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Abstract | References | Similar Articles | Additional Information

Abstract: Polynomials $ {\pi _k}( \cdot ) = {\pi _k}( \cdot ;d\lambda )$, $ k = 0,1,2, \ldots $, are constructed which are orthogonal with respect to the weight distributions $ d\lambda (t) = {(t/({e^t} - 1))^r}\;dt$ and $ d\lambda (t) = {(1/({e^t} + 1))^r}\;dt$, $ r = 1,2$, on $ (0,\infty )$. Moment-related methods being inadequate, a discretized Stieltjes procedure is used to generate the coefficients $ {\alpha _k},{\beta _k}$ in the recursion formula $ {\pi _{k + 1}}(t) = (t - {\alpha _k}){\pi _k}(t) - {\beta _k}{\pi _{k - 1}}(t)$, $ k = 0,1,2, \ldots $, $ {\pi _0}(t) = 1$, $ {\pi _{ - 1}}(t) = 0$. The discretization is effected by the Gauss-Laguerre and a composite Fejér quadrature rule, respectively. Numerical values of $ {\alpha _k},{\beta _k}$, as well as associated error constants, are provided for $ 0 \leqslant k \leqslant 39$. These allow the construction of Gaussian quadrature formulae, including error terms, with up to 40 points. Examples of n-point formulae, $ n = 5(5)40$, are provided in the supplements section at the end of this issue. Such quadrature formulae may prove useful in solid state physics calculations and can also be applied to sum slowly convergent series.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1985-0771039-1
PII: S 0025-5718(1985)0771039-1
Article copyright: © Copyright 1985 American Mathematical Society