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Zbl 1220.65028
Approximations to di- and tri-logarithms.
(English)
[J] J. Comput. Appl. Math. 202, No. 2, 450-459 (2007). ISSN 0377-0427

Using hypergeometric series, simultaneous approximations for polylogarithms are proposed of the form $r_n(z)=a_n \text{Li}_1(z)-b_n$ and $\widetilde{r}_n(z)= a_n \text{Li}_2(z)-\widetilde{b}_n$ where $a_n$ is a polynomial in $1/z$ and $b_n$ and $\widetilde{b}_n$ are sums of polynomials in $1/z$ and $z/(z-1)$. By analytic continuation, this gives simultaneous approximations to $\text{Li}_1(-1)$ and $\text{Li}_2(-1)$ in which case Apéry-like recurrence relations of order 3 for $a_n, b_n$ and $\widetilde{b}_n$, and hence also for $r_n$ and $\widetilde{r}_n$ are obtained. Two generalizations are given. The first is also including $\widetilde{\widetilde{r}}_n(z)=a_n\text{Li}_3(z)-\widetilde{\widetilde{b}}_n$, giving approximations for $z=1$ to $\zeta(2)$ and $\zeta(3)$, and as before, recurrence relations for the $a_n$, $\widetilde{b}_n$, $\widetilde{\widetilde{b}}_n$, $\widetilde{r}_n$ and $\widetilde{\widetilde{r}}_n$. The second generalization introduces well-poised hypergeometric series, which leads for $z=-1$ to simultaneous approximations to the numbers $\pi^2/12$ and $3\zeta(2)/2$.
MSC 2000:
*65D20 Computation of special functions
33C20 Generalized hypergeometric series
33F10 Symbolic computation of special functions
11J70 Continued fractions and generalizations
11M06 Riemannian zeta-function and Dirichlet L-function

Keywords: polylogarithm; dilogarithm; trilogarithm; zeta function; hypergeometric series; Apéry approximation

Cited in: Zbl 1170.34059

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