Approximations to di- and tri-logarithms.(English)Zbl 1220.65028

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:

 65D20 Computation of special functions and constants, construction of tables 33C20 Generalized hypergeometric series, $${}_pF_q$$ 33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.) 11J70 Continued fractions and generalizations 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
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References:

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