Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions

Abstract

Asymptotic approximations ($n \to \infty$) to the truncation errors $r_n = - \sum_{ν=0}^{\infty} a_ν$ of infinite series $\sum_{ν=0}^{\infty} a_ν$ 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 ${}_2 F_1 (a, b; c; z)$ and the divergent asymptotic inverse power series for the exponential integral $E_1 (z)$ -- 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.