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 $(n\to \infty )$ 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 ${}_{2}{F}_{1}(a,b;c;z)$ 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 $L(s,\chi )$ |

33C20 | Generalized hypergeometric series, ${}_{p}{F}_{q}$ |

33E20 | Functions defined by series and integrals |