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Zbl 1117.65042
Weniger, Ernst Joachim
Asymptotic approximations to truncation errors of series representations for special functions.
(English)
[A] Iske, Armin (ed) et al., Algorithms for approximation. Proceedings of the 5th international conference, Chester, UK, July 17--21, 2005. Berlin: Springer. 331-348 (2007). ISBN 3-540-33283-9/hbk

Summary: Asymptotic approximations $(n\to\infty)$ to the truncation errors $r_n=-\sum^\infty_{\nu=n+1}a_\nu$ of infinite series $\sum^\infty_{\nu=0}a_\nu$ 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 $_2F_1(a,b;c;z)$ 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.
MSC 2000:
*65D20 Computation of special functions
33F05 Numerical approximation of special functions
11M06 Riemannian zeta-function and Dirichlet L-function
33C20 Generalized hypergeometric series
33E20 Functions defined by series and integrals

Keywords: symbolic computation; Asymptotic approximations; special functions; Dirichlet series; Riemann zeta function; Euler-Maclaurin formula; Gaussian hypergeometric series; asymptotic inverse power series; exponential integral; numerical examples

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