*(English)*Zbl 1220.65028

Using hypergeometric series, simultaneous approximations for polylogarithms are proposed of the form ${r}_{n}\left(z\right)={a}_{n}{\text{Li}}_{1}\left(z\right)-{b}_{n}$ and ${\tilde{r}}_{n}\left(z\right)={a}_{n}{\text{Li}}_{2}\left(z\right)-{\tilde{b}}_{n}$ where ${a}_{n}$ is a polynomial in $1/z$ and ${b}_{n}$ and ${\tilde{b}}_{n}$ 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 ${\tilde{b}}_{n}$, and hence also for ${r}_{n}$ and ${\tilde{r}}_{n}$ are obtained.

Two generalizations are given. The first is also including ${\tilde{\tilde{r}}}_{n}\left(z\right)={a}_{n}{\text{Li}}_{3}\left(z\right)-{\tilde{\tilde{b}}}_{n}$, giving approximations for $z=1$ to $\zeta \left(2\right)$ and $\zeta \left(3\right)$, and as before, recurrence relations for the ${a}_{n}$, ${\tilde{b}}_{n}$, ${\tilde{\tilde{b}}}_{n}$, ${\tilde{r}}_{n}$ and ${\tilde{\tilde{r}}}_{n}$. The second generalization introduces well-poised hypergeometric series, which leads for $z=-1$ to simultaneous approximations to the numbers ${\pi}^{2}/12$ and $3\zeta \left(2\right)/2$.

##### MSC:

65D20 | Computation of special functions, construction of tables |

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

33F10 | Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.) |

11J70 | Continued fractions and generalizations |

11M06 | $\zeta \left(s\right)$ and $L(s,\chi )$ |