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Asymptotic approximations to truncation errors of series representations for special functions. (English) Zbl 1117.65042
Iske, Armin (ed) et al., Algorithms for approximation. Proceedings of the 5th international conference, Chester, UK, July 17–21, 2005. Berlin: Springer (ISBN 3-540-33283-9/hbk). 331-348 (2007).
Summary: Asymptotic approximations $\left(n\to \infty \right)$ to the truncation errors ${r}_{n}=-{\sum }_{\nu =n+1}^{\infty }{a}_{\nu }$ of infinite series ${\sum }_{\nu =0}^{\infty }{a}_{\nu }$ 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 ${\Delta }{r}_{n}={a}_{n+1}$. 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 ${}_{2}{F}_{1}\left(a,b;c;z\right)$ and the divergent asymptotic inverse power series for the exponential integral ${E}_{1}\left(z\right)$ – 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.
##### MSC:
 65D20 Computation of special functions, construction of tables 33F05 Numerical approximation and evaluation of special functions 11M06 $\zeta \left(s\right)$ and $L\left(s,\chi \right)$ 33C20 Generalized hypergeometric series, ${}_{p}{F}_{q}$ 33E20 Functions defined by series and integrals
Maple