Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions

2005/11/03 by Ernst Joachim Weniger, Weniger, Ernst Joachim
Mathematics · Physics and Astronomy · #11M06 #33C05 #33C70 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Numerical methods for differential equations #Scientific Research and Discoveries

paper · pdf · doi:10.48550/arxiv.math/0511074

openalex publication_date 2005/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Asymptotic approximations (n → ∞) to the truncation errors r_n = - ∑_ν=0^∞ a_ν of infinite series ∑_ν=0^∞ a_ν for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation Δr_n = a_n+1. In the case of the remainder of the Dirichlet series for the Riemann zeta function, the linear equations can be solved in closed form, reproducing the corresponding Euler-Maclaurin formula. In the case of the other series considered -- the Gaussian hypergeometric series ₂ F₁ (a, b; c; z) and the divergent asymptotic inverse power series for the exponential integral E₁ (z) -- the corresponding linear equations are solved symbolically with the help of Maple. The practical usefulness of the new formalism is demonstrated by some numerical examples.

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