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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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