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

Abstract

Polynomials π k ( ⋅ ) = π k ( ⋅ ; d λ ) {\pi _k}( \cdot ) = {\pi _k}( \cdot ;d\lambda ) , k = 0 , 1 , 2 , … k = 0,1,2, \ldots , are constructed which are orthogonal with respect to the weight distributions d λ ( t ) = ( t / ( e t − 1 ) ) r d t d\lambda (t) = {(t/({e^t} - 1))^r}\;dt and d λ ( t ) = ( 1 / ( e t + 1 ) ) r d t d\lambda (t) = {(1/({e^t} + 1))^r}\;dt , r = 1 , 2 r = 1,2 , on ( 0 , ∞ ) (0,\infty ) . Moment-related methods being inadequate, a discretized Stieltjes procedure is used to generate the coefficients α k , β k {\alpha _k},{\beta _k} in the recursion formula π k + 1 ( t ) = ( t − α k ) π k ( t ) − β k π k − 1 ( t ) {\pi _{k + 1}}(t) = (t - {\alpha _k}){\pi _k}(t) - {\beta _k}{\pi _{k - 1}}(t) , k = 0 , 1 , 2 , … k = 0,1,2, \ldots , π 0 ( t ) = 1 {\pi _0}(t) = 1 , π − 1 ( t ) = 0 {\pi _{ - 1}}(t) = 0 . The discretization is effected by the Gauss-Laguerre and a composite Fejér quadrature rule, respectively. Numerical values of α k , β k {\alpha _k},{\beta _k} , as well as associated error constants, are provided for 0 ⩽ k ⩽ 39 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 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.